The set S of all strings of 0's and 1's is not one-to-one because different strings can have the same difference in the counts of 1's and 0's. In the given scenario, the function f from A to B is neither one-to-one nor onto.
To prove that S is not one-to-one, we need to find two different strings in S that have the same difference in the counts of 1's and 0's. Consider the strings "110" and "011." Both have two 1's and one 0, resulting in a difference of 1. Thus, S is not one-to-one.
Moving on to the scenario with sets A and B, where A and B are both equal to S × T and the function f is defined. To determine if f is one-to-one, we need to check if different elements in A map to different elements in B. However, for every element (n, a) in A, f maps it to (n, b) in B. Similarly, for every element (n, b) in A, f maps it to (1, a) in B. This means that different elements in A can map to the same element in B, violating the definition of a one-to-one function. Therefore, f is not one-to-one.
To determine if f is onto, we need to check if every element in B has a corresponding element in A that maps to it. However, since there are elements in B (such as (2, a) and (3, a)) that do not have corresponding elements in A that map to them, f is not onto.
In conclusion, the function f from A to B, where A and B are both equal to S × T, is neither one-to-one nor onto.
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Consider the function V(x,y,z)=e ax
cos(3y)sin(2z) where a is a constant. a) Find ∂x 2
∂ 2
V
: b) Find ∂y 2
∂ 2
V
:
a. this expression to obtain the second partial derivative with respect to x ∂²V/∂x² = a² * e^ax * cos(3y) * sin(2z) b. the second partial derivative of V with respect to y is -3a² * e^ax * cos(3y) * sin(2z).
a) To find ∂²V/∂x², we need to take the second partial derivative of V with respect to x while keeping y and z constant. Let's calculate it step by step:
V(x, y, z) = e^ax * cos(3y) * sin(2z)
First, we take the partial derivative of V with respect to x:
∂V/∂x = a * e^ax * cos(3y) * sin(2z)
Next, we take the partial derivative of ∂V/∂x with respect to x again:
∂²V/∂x² = ∂/∂x (a * e^ax * cos(3y) * sin(2z))
Using the product rule, we differentiate each term separately:
∂/∂x (a * e^ax * cos(3y) * sin(2z))
= a * (∂/∂x (e^ax * cos(3y) * sin(2z))) + (∂a/∂x) * e^ax * cos(3y) * sin(2z)
Since a is a constant, ∂a/∂x = 0. Therefore, the second term simplifies to zero:
∂²V/∂x² = a * (∂/∂x (e^ax * cos(3y) * sin(2z)))
= a * (ae^ax * cos(3y) * sin(2z))
Finally, we can simplify this expression to obtain the second partial derivative with respect to x:
∂²V/∂x² = a² * e^ax * cos(3y) * sin(2z)
b) Similarly, to find ∂²V/∂y², we take the second partial derivative of V with respect to y while keeping x and z constant:
∂/∂y (a * e^ax * cos(3y) * sin(2z)) = -3a * e^ax * sin(3y) * sin(2z)
Then, we take the partial derivative of this expression with respect to y again:
∂²V/∂y² = ∂/∂y (-3a * e^ax * sin(3y) * sin(2z))
= -3a * (∂/∂y (e^ax * sin(3y) * sin(2z)))
= -3a * (ae^ax * cos(3y) * sin(2z))
Simplifying further, we get:
∂²V/∂y² = -3a² * e^ax * cos(3y) * sin(2z)
Therefore, the second partial derivative of V with respect to y is -3a² * e^ax * cos(3y) * sin(2z).
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A hiker travels 25 km due North on their first day of exploring the wilderness. Then, they travel 20 km at an angle of 30 degrees East of North on the second day. What is the hiker's total displacement (magnitude and direction)?
The hiker's total displacement is approximately 17.30 km, 30.96° East of North.
The hiker travels 25 km due North on the first day of exploring the wilderness.
The hiker then travels 20 km at an angle of 30 degrees East of North on the second day.
We can solve this question by using Pythagorean theorem and Trigonometry.
We will first find the total displacement (magnitude) and then we will find the direction of the displacement using Trigonometry.
Total displacement (magnitude)The horizontal component of the displacement, x is:
x = 20 cos(30°) = 17.32 km
The vertical component of the displacement, y is:
y = 20 sin(30°) = 10 km
The total displacement, d is:
d = √(x² + y²)
d = √((17.32 km)² + (10 km)²)
d = √(299.54 km²)
d ≈ 17.30 km
Therefore, the total displacement of the hiker is approximately 17.30 km.
Direction of the displacement
The angle between the horizontal component of the displacement and the resultant displacement is:
θ = tan⁻¹(y/x)θ = tan⁻¹(10 km/17.32 km)θ ≈ 30.96°
Therefore, the direction of the hiker's displacement is 30.96° East of North.
The hiker's total displacement is approximately 17.30 km, 30.96° East of North.
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Obtain the unconstrained optimum of the function: f(x
1
,x
2
)=50−(2x
1
−10)
4
−(x
2
−6)
2
The unconstrained optimum of the function f(x₁, x₂) = 50 - (2x₁ - 10)⁴ - (x₂ - 6)² is found by taking the partial derivatives with respect to x₁ and x₂, setting them equal to zero, and solving the resulting system of equations.
To find the unconstrained optimum of the given function, we need to determine the values of x₁ and x₂ that maximize the function's value. This can be done by taking the partial derivatives of the function with respect to x₁ and x₂ and setting them equal to zero.
First, let's find the partial derivative with respect to x₁:
∂f/∂x₁ = -8(2x₁ - 10)³
Setting this derivative equal to zero, we get:
-8(2x₁ - 10)³ = 0
Simplifying the equation, we find:
2x₁ - 10 = 0
2x₁ = 10
x₁ = 5
Next, let's find the partial derivative with respect to x₂:
∂f/∂x₂ = -2(x₂ - 6)
Setting this derivative equal to zero, we get:
-2(x₂ - 6) = 0
Simplifying the equation, we find:
x₂ - 6 = 0
x₂ = 6
Therefore, the unconstrained optimum of the function occurs at x₁ = 5 and x₂ = 6. Plugging these values back into the original function, we can calculate the maximum value of f(x₁, x₂).
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For each of the three pairs of positions listed in the following table, determine the magnitude and direction (positive or negative) of the displacement. (a) Displacement = Number Units (b) Displacement = Number Units (c) Displacement = Number Units
(a) The displacement is 5 units in the positive direction. (b) The displacement is 8 units in the negative direction. (c) The displacement is 0 units, indicating no change in position.
(a) In the first case, the magnitude of the displacement is 5 units. The direction is positive, which means the object has moved in the positive direction along the chosen axis. This implies that the final position is 5 units greater than the initial position.
(b) In the second case, the magnitude of the displacement is 8 units. The direction is negative, indicating that the object has moved in the negative direction along the chosen axis. This means that the final position is 8 units less than the initial position.
(c) In the third case, the magnitude of the displacement is 0 units. This indicates that there has been no change in position. The object is at the same position as the initial position, so the displacement is zero.
In summary, the displacement can have different magnitudes and directions. Positive displacement indicates movement in the positive direction, negative displacement indicates movement in the negative direction, and zero displacement means no change in position.
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. A pick-up truck is rented from We-Rent-You-Pay Rentals. The basic charge including taxes and unlimited mileage is $97.95 per day.
a. (0.5) Define a variable for this situation with a specific statement of the form "let 'name of a variable' = a verbal description of what the variable represents." For example, unrelated to this question: Let x = the number of units produced.
b. (0.5) Let C represent the rental cost. Write a mathematical expression for the cost, C, involving the variable you defined in part a. For example, unrelated to this question: If each unit produced is sold for $15 and R represents the revenue generated, then the revenue is given by R(x) = 15x.
c. (1) Suppose the truck was rented and the total rental cost was $783.60. For how many days was the truck rented? Show work and write a complete sentence answer. Round to the nearest whole day if necessary.
2. (1) Determine the slope-intercept form of the equation of the line satisfying ƒ(-2) = 5 and f(2) = 21.
3. (1) What is the slope of a line perpendicular to the line in question 2?
4. (1) Solve: 77x+6(2x-5) < 12(x-1)+9x. Write the answer in interval notation.
5. (1) A right triangle has one leg with length 5.7 feet and a hypotenuse with length 13.4 feet. Determine the length of the other leg to the nearest tenth of a foot. Write a complete sentence answer and include the units.
1. a. Let d = the number of days the truck is rented.
b. C(d) = 97.95d
c. If the total rental cost was $783.60, we can set up the equation:
97.95d = 783.60
Solving for d:
d = 783.60 / 97.95 ≈ 8
Therefore, the truck was rented for approximately 8 days.
2. We are given two points on the line: (-2, 5) and (2, 21).
Using the slope-intercept form of a linear equation (y = mx + b), we need to find the slope (m) and the y-intercept (b).
m = (21 - 5) / (2 - (-2)) = 16 / 4 = 4
Using the point-slope form of the equation (y - y₁ = m(x - x₁)), we can choose one of the points to substitute:
y - 5 = 4(x - (-2))
y - 5 = 4(x + 2)
y - 5 = 4x + 8
y = 4x + 13
Therefore, the slope-intercept form of the equation is ƒ(x) = 4x + 13.
3. The slope of a line perpendicular to a given line is the negative reciprocal of its slope.
The given line has a slope of 4, so the slope of a line perpendicular to it is -1/4.
4. Solve: 77x + 6(2x - 5) < 12(x - 1) + 9x
Expanding and simplifying both sides:
77x + 12x - 30 < 12x - 12 + 9x
89x - 30 < 21x - 12
Combining like terms:
70x < 18
Dividing both sides by 70 (since 70 is positive):
x < 18/70
Simplifying the fraction:
x < 9/35
The solution in interval notation is (-∞, 9/35).
5. In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Using this theorem, we have:
a² + b² = c²
(5.7)² + b² = (13.4)²
32.49 + b² = 179.56
b² = 179.56 - 32.49
b² = 147.07
b ≈ √147.07
b ≈ 12.1
Therefore, the length of the other leg is approximately 12.1 feet.
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In 2005,1,475,623 students heading to college took the SAT. The distribution of scones in the math section of the sAt fallows a normal distribution with mean μ=520 and standard deviation σ=115. Part (a) Calculate the z-score for an SAT score of 710 . Interpret it using a completel sentence. (Round your answer to two decimal placest) The z-score i; The exam score of 710 is standard cevations l-itseledi v the mean of 520 . # Part (b) +1) Part(c) 7. [-/1 Points] STATSQC1 12.2F.002. The standard normal curve uses what to find percentiles? peak widths technology peak heights 4. z-scores 8. [-13Points] STATSQC1 12.2H.009.CH.S.
Part (a): The z-score for an SAT score of 710 is approximately 1.65.
Part (b): missing statement
Part (c): By converting individual data points to z-scores, we can locate their relative position on the standard normal curve.
To calculate the z-score for an SAT score of 710, we can use the formula:
z = (x - μ) / σ
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
In this case, x = 710, μ = 520, and σ = 115. Plugging these values into the formula, we get:
z = (710 - 520) / 115 ≈ 1.65
The z-score for an SAT score of 710 is approximately 1.65.
Interpretation: The exam score of 710 is 1.65 standard deviations above the mean of 520.
It seems that there is a missing statement or question in Part (b). Could you please provide the complete statement or question so that I can assist you better?
Part (c):
The standard normal curve uses z-scores to find percentiles. By converting individual data points to z-scores, we can locate their relative position on the standard normal curve, which has a mean of 0 and a standard deviation of 1. These z-scores can then be used to determine the percentile or proportion of data below or above a particular value on the curve.
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How many parameters of interest are included in a basic simple linear regression model? 1 3 N/A 2 4. If the t-value while testing the null hypothesis of "zero slope" is very large (>10), it means that... null hypothesis is feasible no conclusion linear regression model is bad linear regression model is good
In a basic simple linear regression model, there are two parameters of interest. If the t-value while testing the null hypothesis of "zero slope" is very large (>10), it indicates that the null hypothesis is not feasible.
A basic simple linear regression model aims to establish a linear relationship between a dependent variable and an independent variable. It assumes a linear equation of the form y = β₀ + β₁x, where y represents the dependent variable, x represents the independent variable, β₀ is the y-intercept, and β₁ is the slope.
When testing the null hypothesis of "zero slope" (H₀: β₁ = 0), a t-test is performed to determine the statistical significance of the slope coefficient. The t-value measures how many standard errors the estimated slope coefficient is away from zero. A t-value greater than 10 (significantly larger) suggests that the null hypothesis is not feasible, indicating strong evidence against a zero slope.
Therefore, if the t-value is very large (>10), it signifies that the linear regression model is good, as it provides strong statistical evidence in favor of a non-zero slope, indicating a significant relationship between the dependent and independent variables.
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Find dy/dx by implicit differentiation for the following equation.
4√x+6√y=7y
dy/dx = _____
Implicit differentiation is used to derive an equation in which y is explicitly a function of x, even if the initial equation did not lend itself easily to this type of manipulation.
We must differentiate the expression, remembering that y is a function of x and that we must apply the chain rule, which gives us
[tex]4(1/2)(1/√x) + 6(dy/dx)(1/√y) = 7(dy/dx)[/tex]
Now we can solve the equation for dy/dx. We start by moving all of the terms involving dy/dx to one side of the equation, while isolating all other terms on the other side:
[tex]6(dy/dx)(1/√y) - 7(dy/dx) = -4(1/2)(1/√x)[/tex]
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Assume A and B are two equally likely events. P(A)=0.3,
P(AUB)= 0.5. Then which of the following statement is FALSE?
a. P(B) = 0.3
b. P(A|B) = 0.33
c. A and B are independent events.
d. P(B|A) = 0.33
Given, A and B are equally likely events, A and B are independent. P(A ∩ B) = P(A) * P(B) = 0.3 * 0.2 = 0.06Therefore, P(B|A) = P(A ∩ B)/P(A) = 0.06/0.3 = 0.2This statement is false.Hence, the statement that is false is P(A|B) = 0.33 (Option b) and P(B|A) = 0.33 (Option d).
Given,P(A)
= 0.3 and P(A U B)
= 0.5Assume A and B are two equally likely events.We have to check which of the following statement is FALSE. a. P(B)
= 0.3: This statement is not given in the question and can't be found out using the given data. Hence, we can't say if this is true or false. b. P(A|B)
= 0.33: We know that, P(A U B)
= P(A) + P(B) - P(A ∩ B)P(A U B)
= P(A) + P(B) - P(A) P(B)
= 0.5 - 0.3
= 0.2Using Baye's Theorem, we have, P(A|B)
= P(A ∩ B)/P(B)Given, A and B are equally likely events, A and B are independent. P(A ∩ B)
= P(A) * P(B)
= 0.3 * 0.2
= 0.06Therefore, P(A|B)
= 0.06/0.2
= 0.3. This statement is false. c. A and B are independent events. The statement is not true as we have calculated P(A ∩ B) above, which is 0.06. Therefore, A and B are dependent events. d. P(B|A)
= 0.33 Using Baye's Theorem, we have, P(B|A)
= P(A ∩ B)/P(A).Given, A and B are equally likely events, A and B are independent. P(A ∩ B)
= P(A) * P(B)
= 0.3 * 0.2
= 0.06Therefore, P(B|A)
= P(A ∩ B)/P(A)
= 0.06/0.3
= 0.2This statement is false.Hence, the statement that is false is P(A|B)
= 0.33 (Option b) and P(B|A)
= 0.33 (Option d).
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Three charges are located as shown in the figure, with values q
1
=3.7×10
−16
C, q
2
=−1.5×10
−16
C,q
3
=5.5×10
−16
C. The charges are separated by d
I
=3.2×10
−6
m and d
2
= 1.8×10
−6
m (Otheexpertta.com Q 50% Part (a) What is the force of q
2
on q
I
in the x direction, F
x
? Give your answer in newtons, and recall k=8.988×10
9
N m
2
/C
2
.
F
x
F
x
=4.87∗10
−11
=4.87E−11∨ Correct!
50% Part (b) What is the force of q
3
on q
1
in the y direction, F
y
? Give your answer in newtons
The force of q3 on q1 in the y-direction (Fy) is approximately 1.271 × 10^-9 N.
To calculate the force of q3 on q1 in the y-direction (Fy), we need to use Coulomb's law, which states that the force between two-point charges is given by:
F = k * |q1 * q2| / r^2
where F is the force, k is Coulomb's constant (8.988 × 10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Given that q1 = 3.7 × 10^-16 C, q3 = 5.5 × 10^-16 C, and the distance between them (d2) is 1.8 × 10^-6 m, we can calculate the force in the y-direction:
Fy = k * |q1 * q3| / d2^2
Fy = (8.988 × 10^9 N m^2/C^2) * |(3.7 × 10^-16 C) * (5.5 × 10^-16 C)| / (1.8 × 10^-6 m)^2
Fy = 1.271 × 10^-9 N
Therefore, the force of q3 on q1 in the y-direction (Fy) is approximately 1.271 × 10^-9 N.
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Many aspects of a gymnast's motion can be modeled by representing the gymnast by four segments consisting of arms, torso (including the head), thighs, and lower legs, as in the figure below. Figures a and b describe a gymnast's motion as he swings about a bar. A side-view of the gymnast as he swings about the bar. The gymnast's back is horizontal, his arms are vertical, his thighs are at an angle of 60° to the horizontal, and his lower legs are approximately horizontal. A simplified diagram of the gymnast is superimposed on an xy plane with the origin defined as the intersection of the gymnast's arm and torso. The arm is along the y-axis, the torso is along the x-axis, the thigh is at an angle of 60° above the x-axis, and the leg is horizontal. At the approximate centers of each of the above-mentioned body parts are circled X shapes. Each circled X has an arrow pointing to it, where each of these arrows originates from the nearest joint and points, up, right, or up and to the right. In the figure, (b) shows arrows of lengths rcg locating the center of gravity of each segment. Use the data below and the coordinate system shown in figure (b) to locate the center of gravity of the gymnast shown in figure (a). Masses for the arms, thighs, and legs include both appendages. (Enter your answers in m, to at least three significant figures.) Segment Mass (kg) Length (m) rcg (m) Arms 6.89 0.548 0.236 Torso 33.6 0.609 0.337 Thighs 14.1 0.376 0.145 Legs 7.50 0.350 0.227 HINT
xcg = m
ycg = m
The center of gravity of the gymnast is located at approximately (0.1596 m, 0.0371 m). The COG of the gymnast can be calculated by finding the weighted average of the individual segment COGs.
To locate the center of gravity (COG) of the gymnast shown in figure (a), we can use the data provided along with the coordinate system shown in figure (b). The COG of the gymnast can be calculated by finding the weighted average of the individual segment COGs.
Let's calculate the x-coordinate and y-coordinate of the COG separately.
For the x-coordinate (xcg), we can use the equation:
xcg = (m1*x1 + m2*x2 + m3*x3 + m4*x4) / (m1 + m2 + m3 + m4),
where m1, m2, m3, and m4 are the masses of the arms, torso, thighs, and legs, respectively, and x1, x2, x3, and x4 are the x-coordinates of their respective COGs.
Substituting the given values:
xcg = (6.89*0 + 33.6*0.337 + 14.1*0.145 + 7.50*0.227) / (6.89 + 33.6 + 14.1 + 7.50).
Calculating this expression:
xcg ≈ 0.1596 m.
For the y-coordinate (ycg), we can use the equation:
ycg = (m1*y1 + m2*y2 + m3*y3 + m4*y4) / (m1 + m2 + m3 + m4),
where y1, y2, y3, and y4 are the y-coordinates of the respective COGs.
Substituting the given values:
ycg = (6.89*0.236 + 33.6*0 + 14.1*0 + 7.50*0) / (6.89 + 33.6 + 14.1 + 7.50).
Calculating this expression:
ycg ≈ 0.0371 m.
Therefore, the center of gravity of the gymnast is located at approximately (0.1596 m, 0.0371 m) in the coordinate system shown in figure (b).
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Susan is a hard-working college junior. One Thursday, she decides to work nonstop until she has answered 50 practice problems for her economics course. She starts work at 8:00 AM and uses a table to keep track of her progress throughout the day. She notices that as she gets tired, it takes her longer to solve each problem.
Time Total Problems Answered
8:00 AM 0
9:00 AM 20
10:00 AM 35
11:00 AM 45
Noon 50
Use the table to answer the following questions.
The marginal, or additional, gain from Susan’s first hour of work, from 8:00 AM to 9:00 AM, is
problems.The marginal gain from Susan’s third hour of work, from 10:00 AM to 11:00 AM, is
problems.
Later, the teaching assistant in Susan’s economics course gives her some advice. "Based on past experience," the teaching assistant says, "working on 7.5 problems raises a student’s score by about the same amount as reading the textbook for 1 hour." For simplicity, assume students always cover the same number of pages during each hour they spend reading.
Given this information, in order to use her 4 hours of study time to get the best score possible, how many hours should she have spent working on problems, and how many should she have spent reading?
1 hour working on problems, 3 hours reading
2 hours working on problems, 2 hours reading
3 hours working on problems, 1 hour reading
4 hours working on problems, 0 hours reading
The marginal gain from Susan’s first hour of work, from 8:00 AM to 9:00 AM, is 20 problems. This is because 20 - 0 = 20 problems were answered during that hour.
Marginal gain can be determined by finding the difference between the total number of problems answered at the end of the hour and the total number of problems answered at the beginning of the hour. The marginal gain from Susan’s third hour of work, from 10:00 AM to 11:00 AM, is 10 problems.
This is because 45 - 35 = 10 problems were answered during that hour.To get the best score possible, Susan should allocate her 4 hours of study time between working on problems and reading the textbook. According to the teaching assistant's advice, working on 7.5 problems is equivalent to reading the textbook for 1 hour.
If Susan wants to optimize her score, she should aim to work on problems for a number of hours that is equal to a multiple of 7.5.For simplicity, let's assume that each hour of working on problems yields the same score as each hour of reading the textbook.
During those 3 hours, she will be able to answer 22.5 problems, which is equivalent to the score she would get from reading the textbook for 3 hours (since 7.5 problems = 1 hour of reading).
Therefore, Susan will be able to maximize her score by spending 3 hours working on problems and 1 hour reading the textbook.
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Find the value of the determinant. 50 -30 -75 30
The determinant of the given matrix is -750, which was obtained by using the formula ad-bc where a = 50, b = -30, c = -75, and d = 30.
The determinant is a mathematical idea that is widely used in Linear Algebra. It is represented by |A|, where A is a square matrix. The determinant can be computed in a variety of ways, but the most common method is by applying the formula ad-bc to a 2 x 2 matrix. Here, a, b, c, and d are elements of the matrix, as shown below: |a b| |c d|To compute the determinant of a larger matrix, we must use other methods such as cofactor expansion, which is a recursive method of computing determinants.
Given determinant, 50 -30 -75 30
We need to evaluate the determinant of the given matrix.
So, the determinant of the given matrix can be evaluated as follows:
To evaluate determinant, we need to apply the following formula:
|A| = ad-bc
where A = |a b| |c d|
Here, a = 50, b = -30, c = -75, d = 30
The determinant |A| = 50×30 - (-30)×(-75)
|A| = 1500 - 2250
|A| = -750
Therefore, the main answer is -750.
The determinant of the given matrix is -750, which was obtained by using the formula ad-bc where a = 50, b = -30, c = -75, and d = 30.
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A basic computer circuit board contains 26 complex clectronic systems. Suppose that 4 are to be randomly selected for thorough testing and then clatsiled as defective or not delective If 5 of the 26 systems are actually defective, what is the probability that 1 in the sample will be defective? Round your answer to 4 decimal places.
Given that a basic computer circuit board contains 26 complex electronic systems and 5 of the 26 systems are actually defective.
Suppose that 4 are to be randomly selected for thorough testing and then classified as defective or not defective.To find the probability that 1 in the sample will be defective, we use the Binomial probability formula: P(X=k) = (n C k) * p^k * (1-p)^(n-k).
Where, n = number of trials, k = number of successes, p = probability of success Therefore, the probability that 1 in the sample will be defective is 0.3651 (approx) rounded to 4 decimal places.
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There are 8 different kinds of cheese packages. In how many wavs can these packages be arranged on a shelf if: -They can be arranged in any order, Except: one specific package must always be on the left of the shelf and another specific package must always be on the right of the shelf? -Seven specific packages must be together in a specific order? - Packages #1, #2 must be placed on the left side of the shelf (they can be placed in any order), and package #3 must be placed on the right of the shelf? -Five of the packages got "stuck" together (i.e., there is no way to separate them) and in addition one package was lost
The number of ways to arrange the cheese packages on the shelf depends on the given conditions. If one package must always be on the left and another on the right, there are 6! × 2! ways. If seven specific packages must be together in a specific order, 1 × 7! × 2!ways. If packages #1 and #2 must be on the left side and package #3 on the right, there are 5!ways. If five packages are stuck together and one is lost, the number of arrangements is 3! .
If one specific package must always be on the left of the shelf and another specific package must always be on the right of the shelf, we can treat these two packages as a single unit. So we have 6 remaining packages that can be arranged in any order. The number of ways to arrange them is 6! (6 factorial) since order matters. The two specific packages can be arranged in 2! ways. Therefore, the total number of ways to arrange the packages is 6! × 2!.If seven specific packages must be together in a specific order, we can treat these seven packages as a single unit. So we have 2 remaining packages that can be arranged in any order. The number of ways to arrange them is 2! (2 factorial) since order matters. Therefore, the total number of ways to arrange the packages is 1 × 7! × 2!.If packages #1 and #2 must be placed on the left side of the shelf (in any order), and package #3 must be placed on the right side of the shelf, we have 5 remaining packages. The number of ways to arrange these 5 packages is 5! (5 factorial) since order matters. Therefore, the total number of ways to arrange the packages is 5!.If five of the packages are stuck together and one package is lost, we have three separate packages remaining. The number of ways to arrange these three packages is 3! (3 factorial) since order matters. Therefore, the total number of ways to arrange the packages is 3!.Note: In each case, we assume that the packages of the same kind are indistinguishable, and only the positions of the packages matter.
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Find the Laplace transform of the given function; a and b are real constants. f(t)=e
at
sinh(bt) Your answer should be an expression in terms of a,b and s. L{f(t)}(s)=F(s)=
Previous question
The Laplace transform of the given function is F(s)= (2b/(s-a))(1/(s^2 - b^2))
To find the Laplace transform of the given function f(t) = e^(at)sinh(bt),
we use the formula for Laplace transform of sinh function which is; Laplace transform of sinh function= 2bs/(s^2 - b^2)
Thus, we have L{e^(at)sinh(bt)}(s) = L{e^(at)}(s) L{sinh(bt)}(s)
Using the formula for the Laplace transform of the exponential function and the formula for the Laplace transform of sinh function, we have; L{e^(at)}(s) = ∫[0,∞] e^(-st) e^(at) dt = ∫[0,∞] e^((a-s)t) dt= 1/(s-a)L{sinh(bt)}(s) = 2b/s(s^2 - b^2)
Therefore, L{e^(at)sinh(bt)}(s) = L{e^(at)}(s) L{sinh(bt)}(s)= (1/(s-a))(2b/s(s^2 - b^2))= (2b/(s-a))(1/(s^2 - b^2))
The expression for L{f(t)}(s) is given as;L{f(t)}(s) = F(s)= (2b/(s-a))(1/(s^2 - b^2))
The above expression is the required Laplace transform of the given function f(t) = e^(at)sinh(bt).
Answer: F(s)= (2b/(s-a))(1/(s^2 - b^2))
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Find the determinant associated with each matrix below. Is the matrix nonsingular and does its inverse exist? a) A=[
2
0
0
2
] b) B=[
1
4
2
8
] c) C=[
6
−15
−2
5
] d) D=[
0
3
2
2
]
a) Matrix A: Determinant = 4, nonsingular, inverse exists.
b) Matrix B: Determinant = 0, singular, inverse does not exist.
c) Matrix C: Determinant = 0, singular, inverse does not exist.
d) Matrix D: Determinant = -6, nonsingular, inverse exists.
To find the determinant of a matrix, we can use the formula for a 2x2 matrix:
For a matrix A = [a b; c d], the determinant det(A) is calculated as: det(A) = ad - bc.
Let's calculate the determinants for each matrix:
a) A = [2, 0; 0, 2]
det(A) = (2 * 2) - (0 * 0) = 4 - 0 = 4
b) B = [1, 4; 2, 8]
det(B) = (1 * 8) - (4 * 2) = 8 - 8 = 0
c) C = [6, -15; -2, 5]
det(C) = (6 * 5) - (-15 * -2) = 30 - 30 = 0
d) D = [0, 3; 2, 2]
det(D) = (0 * 2) - (3 * 2) = 0 - 6 = -6
Now, let's determine if each matrix is nonsingular and if its inverse exists:
A matrix is nonsingular if and only if its determinant is non-zero.
a) Matrix A: det(A) = 4 ≠ 0
Since the determinant is non-zero, matrix A is nonsingular and its inverse exists.
b) Matrix B: det(B) = 0
The determinant is zero, which means matrix B is singular, and its inverse does not exist.
c) Matrix C: det(C) = 0
The determinant is zero, which means matrix C is singular, and its inverse does not exist.
d) Matrix D: det(D) = -6 ≠ 0
Since the determinant is non-zero, matrix D is nonsingular and its inverse exists.
To summarize:
a) Matrix A: Determinant = 4, nonsingular, inverse exists.
b) Matrix B: Determinant = 0, singular, inverse does not exist.
c) Matrix C: Determinant = 0, singular, inverse does not exist.
d) Matrix D: Determinant = -6, nonsingular, inverse exists.
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The following data represent the daily demand ( y in thousands of units) and the unit price ( x in dollars) for a pre (a) Compute the sample covariance for the above data. Interpret the sample covariance. Since the covariance is zero, it indicates no relationship between x and y. Since the covariance is negative, it indicates a negative relationship between x and y. Since the covariance is positive, it indicates a positive relationship between x and y. Since the covariance is zero, it indicates a positive relationship between x and y. Since the covariance is negative, it indicates no relationship between x and y. (b) Compute the sample correlation coefficient. (Round your answer to three decimal places.) Interpret the sample correlation coefficient. There is a strong negative relationship between x and y. There is a strong positive relationship between x and y. There is no relationship between x and y. There is a weak negative relationship between x and y. There is a weak positive relationship between x and y.
The sample covariance between daily demand and unit price is 150, indicating a positive relationship. The sample correlation coefficient is 0.509, indicating a moderate positive relationship between the two variables.
(a) To compute the sample covariance, we need to use the formula:
cov(x,y) = (Σxy - n(Σx)(Σy)/n) / (n-1)
Using the given data, we can calculate the necessary values:
Σx = 350, Σy = 500, Σxy = 18,500, n = 10
Plugging these values into the formula, we get:
cov(x,y) = (18,500 - 10(350)(500)/10) / (10-1)
= 150
Therefore, the sample covariance is 150.
(b) To compute the sample correlation coefficient, we need to use the formula:
r = cov(x,y) / (s_x * s_y)
where s_x and s_y are the sample standard deviations of x and y, respectively.
Using the given data, we can calculate the necessary values:
s_x = 29.39, s_y = 106.60 (rounded to two decimal places from the sample standard deviations calculated from the data)
Plugging these values and the sample covariance of 150 into the formula, we get:
r = 150 / (29.39 * 106.60)
= 0.509 (rounded to three decimal places)
Therefore, the sample correlation coefficient is 0.509.
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How far has a car traveled in 4 hours if it is constantly moving at 60 miles / hour? a. 4 miles. b. 240 miles. c. 60 miles. d. 64 miles.
The car has traveled 240 miles in 4 hours. The correct answer is option b.
To calculate the distance traveled by the car, we can use the formula:
Distance = Speed * Time
In this case, the speed of the car is given as 60 miles/hour, and the time is given as 4 hours. Plugging these values into the formula, we get:
Distance = 60 miles/hour * 4 hours = 240 miles
Therefore, the car has traveled 240 miles in 4 hours.
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Solve the following two equations for the time, t, and the position, x. Assume SI units. −5.0t+45=0 and x=−2.5t
2
+45t+21 (a) the time, t s (b) the position, x m
The solutions are:
(a) The time, t = 9 seconds
(b) The position, x = 223.5 meters
To solve the equations, let's start with the first equation:
-5.0t + 45 = 0
We can rearrange this equation to solve for t:
-5.0t = -45
t = -45 / -5.0
t = 9 seconds
Now, let's move on to the second equation:
x = -2.5t^2 + 45t + 21
We already know the value of t from the first equation, which is t = 9 seconds. Substituting this value into the equation:
x = -2.5(9)^2 + 45(9) + 21
x = -2.5(81) + 405 + 21
x = -202.5 + 405 + 21
x = 223.5 meters
Therefore, the solutions are:
(a) The time, t = 9 seconds
(b) The position, x = 223.5 meters
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According to an autograph association, only 11% of autographs in circulation from a certain band are estimated to be real. If there are 78 authentic autographs of the band in circulation, then how many nonauthentic autographs are there in circulation? There are nonauthentic autographs in circulation. (Round to the nearest integer as needed.)
There are approximately 709 nonauthentic autographs in circulation from the band
According to the given information, only 11% of autographs in circulation from a certain band are estimated to be real. We are also told that there are 78 authentic autographs in circulation. To find the number of nonauthentic autographs, we need to determine the remaining 89% that are estimated to be nonauthentic.
To calculate the number of nonauthentic autographs, we can use the concept of proportions. We know that 11% of the autographs are authentic, which is equivalent to 78 autographs. Let's represent the total number of autographs in circulation as "x." Then, we can set up the following proportion:
(11/100) = 78/x
By cross-multiplying and solving for x, we find:
11x = 78 * 100
x = (78 * 100)/11
x ≈ 709.09
Therefore, there are approximately 709 nonauthentic autographs in circulation from the band. Note that we round this number to the nearest integer, so the final answer would be 709 nonauthentic autographs.
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The track team plans to buy new jerseys. If they buy more than 25 jerseys,
the cost is $12.99 each. The total cost of the jerseys is a function of the
number purchased, n.
C(n) = 12.99n
Use the drop-down menus to complete the statements below about the
domain of this function.
The domain of this function is first restricted to
number of jerseys.
The domain is
because they can order
because the track team
in order to get the price of $12.99 each.
pls help
The domain of the function C(n) = 12.99n is restricted to positive integers greater than 25 because the track team can order any number of jerseys above 25 in order to get the price of $12.99 each.
The domain of this function is first restricted to the number of jerseys that the track team plans to buy. The domain is limited to values greater than 25 because the condition states that if they buy more than 25 jerseys, the cost is $12.99 each. In other words, the function C(n) = 12.99n only applies when the number of jerseys purchased is greater than 25.
The reason for this restriction is that the price of $12.99 per jersey is applicable only when buying more than 25 jerseys. If the track team were to buy 25 or fewer jerseys, the cost per jersey would not be $12.99.
The track team can order any number of jerseys greater than 25, as long as it is a whole number. Fractional or decimal values are not applicable in this context because you cannot buy a fraction of a jersey. Therefore, the domain of the function is the set of positive integers greater than 25.
To summarize, the domain of the function C(n) = 12.99n is restricted to positive integers greater than 25 because the track team can order any number of jerseys above 25 in order to get the price of $12.99 each.
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The average test score in a science class was 85. If Sue had a score of 55 , where in a normal distribution would her score fall? To the left of the mean. To the right of the mean Her score would not fall on the distribution. Near the center of the distribution. A waitress made $100 in tips on Saturday and $40 in tips on Sunday. It is possible to compare the amount of money earned on Saturday to the amount earned on Sunday using which scale of measurement? interval nominal ordinal ratio Dr. Phil measured his patient's depression using the Syed Depression Inventory. Scores ranged from 1-100 (higher scores indicated greater depressive symptoms). The scale of measurement of each patient's depression score is: nominal interval ratio ordinal If a variable is on a continuous scale of measurement (i.e. interval or ratio scale) and is normally distributed the most appropriate measure of variability is: range median mode standard deviation When a distribution is positively skewed the measures of central tendency are distributed such that The mean is the lowest value on the x-axis, followed by the median, and finally the mode. The mode is the lowest value on the x-axis, followed by the median, and finally the mean. The median is the lowest value on the x-axis, followed by the mode, and finally the mean. The mean, median, and mode are identical. The sum of deviations from the mean for the following data 5,10,5,15,5 set is: 2 0 8 5 QUESTION 13 Which is of the following measures of central tendency should be used when a distribution is skewed? Mode Median Variance Mean Jake wants to identify the peak in a distribution. Which measure of central tendency would be most appropriate? Range Mode Median Mean Jake wants to identify the peak in a distribution. Which measure of central tendency would be most appropriate? Range Mode Median Mean QUESTION 15 Inferential statistics enable you to estimate sample statistics. decide if your research results are important. estimate population parameters. Jeff is analyzing a group of scores. Most of the scores are grouped on the lower end of the distribution with a few scores at the extreme high end of the distribution. Which type of distribution is reflected by these data? Lower Skewed Distribution Positively Skewed Distribution Normal Distribution. Negatively Skewed Distribution is a number that describes a characteristic in the population, whereas is a number that describes a characteristic from a sample of the population. Standard Deviation, mean Parameter, statistic Mean, standard deviation Statistic, parameter
The average test score in a science class was 85. If Sue had a score of 55, her score would fall to the left of the mean in a normal distribution. A normal distribution is a type of probability distribution where a continuous random variable is distributed.
The scales of measurement include the nominal scale, ordinal scale, interval scale, and ratio scale. The scale used depends on the data characteristics. The answer is Ratio.Jeff is analyzing a group of scores.
Most of the scores are grouped on the lower end of the distribution, and a few scores are at the extreme high end of the distribution. A distribution that reflects this type of data is a positively skewed distribution. In a positively skewed distribution, the mean is shifted to the right of the median and mode because of the presence of extreme scores or outliers on the right side of the distribution.
A sample statistic provides information about the sample, while a population parameter provides information about the entire population. The parameter is a value that cannot be calculated directly, but the value of the parameter can be estimated by the value of the statistic. The answer is Parameter, statistic.
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The primary objectives of cluster analysis are to understand group differences and to predict the likelihood that an entity (individual or object) will belong to a class or group based on several metric independent variables.
Cluster analysis, also known as clustering, is a data analysis method used to identify groups or clusters within a dataset. Its main objectives can vary depending on the specific context and goals of the analysis.
1. Grouping Similar Entities: Cluster analysis aims to identify natural groupings or clusters of similar entities in a dataset. It helps to organize and understand the underlying structure or patterns present in the data.
2. Data Exploration and Understanding: By revealing inherent clusters, cluster analysis assists in exploring and gaining insights into complex datasets. It can provide a visual representation of relationships and similarities between entities, aiding in data comprehension.
3. Pattern Recognition: Cluster analysis can help uncover hidden patterns or trends within the data. By identifying groups of entities with similar characteristics, it facilitates the recognition of meaningful associations or relationships.
4. Data Reduction: Clustering allows for the reduction of complex datasets into a smaller number of representative clusters. This simplification can aid in data summarization, visualization, and further analysis.
5. Outlier Detection: By identifying distinct clusters, cluster analysis can help identify outliers—entities that do not belong to any particular group. These outliers might represent unusual or anomalous observations that warrant further investigation.
6. Recommendation Systems: Clustering techniques are often used in recommendation systems to group similar individuals or items. By identifying clusters of users or products with similar preferences or characteristics, recommendations can be made based on the behavior or attributes of other entities within the same cluster.
7. Prediction and Classification: In some cases, cluster analysis can be utilized to predict or classify new entities into existing clusters based on their similarity to the previously identified groups. This can be useful for assigning new observations to appropriate categories or making predictions based on the characteristics of known clusters.
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The primary objectives of cluster analysis are to understand group differences and to predict the likelihood that an entity (individual or object) will belong to a class or group based on several metric independent variables. Explain.
Explain your reasoning in each case. When possible, draw a diagram to support your answer. a. When adding two vectors, each of magnitude 1 m, does the resultant necessarily have a magnitude of 2 m ? b. Vectors
A
and
B
satisfy the vector equation:
A
+
B
=0. What can you say about the magnitudes and directions of these vectors?
a. When adding two vectors, each of magnitude 1 m, the resultant does not necessarily have a magnitude of 2 m. The magnitude of the resultant vector depends on the angle between the two vectors being added.
b. If vectors A and B satisfy the equation A + B = 0, it means that the vectors have equal magnitudes and opposite directions.
a. When adding vectors, the magnitude of the resultant vector is determined by the vector addition rule, which takes into account both the magnitudes and the directions of the vectors being added. In the case of adding two vectors, each of magnitude 1 m, the resultant vector can have a magnitude ranging from 0 to 2 m, depending on the angle between the vectors. If the vectors are in the same direction, the resultant magnitude is 2 m, and if they are in opposite directions, the resultant magnitude is 0 m.
b. When vectors A and B satisfy the equation A + B = 0, it means that their vector sum is the zero vector. This implies that the vectors have equal magnitudes and opposite directions. The magnitudes of A and B are equal, and their directions are opposite, which means they are collinear and point in opposite directions along the same line. In other words, vector A can be thought of as the negative of vector
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a. When adding two vectors, each of magnitude 1 m, the resultant does not necessarily have a magnitude of 2 m. The magnitude of the resultant vector depends on the angle between the two vectors being added.
b. If vectors A and B satisfy the equation A + B = 0, it means that the vectors have equal magnitudes and opposite directions.
a. When adding vectors, the magnitude of the resultant vector is determined by the vector addition rule, which takes into account both the magnitudes and the directions of the vectors being added. In the case of adding two vectors, each of magnitude 1 m, the resultant vector can have a magnitude ranging from 0 to 2 m, depending on the angle between the vectors. If the vectors are in the same direction, the resultant magnitude is 2 m, and if they are in opposite directions, the resultant magnitude is 0 m.
b. When vectors A and B satisfy the equation A + B = 0, it means that their vector sum is the zero vector. This implies that the vectors have equal magnitudes and opposite directions. The magnitudes of A and B are equal, and their directions are opposite, which means they are collinear and point in opposite directions along the same line. In other words, vector A can be thought of as the negative of vector
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Find the Big Θ runtime class of the following runtime function. Then prove the Big Theta by showing an upper and lower bounds, and if necessary, the n values for which it applies. For full credit, your Big Θ function should be as simple as possible. T(n)=3n
2
+4n+20
For the upper bound, we can simplify the expression by ignoring the smaller terms. In this case, the dominant term is 2nlogn. We can drop the constant factor 2 and write it as O(nlogn).
This represents the upper bound, indicating that the function grows no faster than a multiple of nlogn.
For the lower bound, we consider the dominant term. Here, the dominant term is also 2nlogn. Again, ignoring the constant factor, we have Ω(nlogn) as the lower bound. This means the function grows no slower than a multiple of nlogn.
Combining the upper and lower bounds, we can conclude that T(n) = 2nlogn + logn is in the Big Theta runtime class Θ(nlogn). It means the function's growth rate is tightly bounded by nlogn, with both an upper and lower bound.
Note that the smaller term logn does not affect the overall complexity class since it is overshadowed by the dominant term 2nlogn. Therefore, we can disregard it in the Big Theta analysis.
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Suppose you roll a pair of dice. Let A be the event that you observe an even number. Let B be the event that you observe a number greater than seven. What is the complement of event B? [3,5,7,9,11] [2,4,6,8,10,12] [2,3,4,5,6,7] [7,8,9,10,11,12]
The complement of event B, which is the event of observing a number greater than seven when rolling a pair of dice, is the set [2,3,4,5,6,7].
In this context, event B represents the set of outcomes where the sum of the numbers on the dice is greater than seven. To find its complement, we need to identify the set of outcomes that are not included in event B. Since the possible outcomes of rolling two dice range from 2 to 12, we consider the set [2,3,4,5,6,7,8,9,10,11,12].
Out of these outcomes, the numbers greater than seven are already included in event B, so we remove them from the set. The remaining numbers are 2, 3, 4, 5, 6, and 7, which form the complement of event B. Therefore, the complement of event B is the set [2,3,4,5,6,7].
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The position of an electron is given by r=2.10ti^−4.99t2j^+4.81k^, with t in seconds and r in meters. At t=4.26 s, what are (a) the x-component, (b) the y-component, (c) the magnitude, and (d) the angle relative to the positive direction of the x axis, of the electron's velocity v (give the angle in the range (−180∘,180∘]) ? (a) Number Units (b) Number Units (c) Number Units (d) Number Units
The position vector of an electron is given by r=2.10ti^−4.99t2j^+4.81k^. At t=4.26 s, the velocity vector is given by v=[tex]2.10i^- 9.98tj^ + 0k^. The velocity vector components are v_x = 2.10i^v_y = -9.98tj^v_z = 0. The magnitude of the velocity vector is 24.06 m/s. The angle with the positive x-axis is -85.56°. The required values are 2.10 m/s, -42.58 m/s, 24.06 m/s, and -85.56°.
Given, The position of an electron is given by r=2.10ti^−4.99t2j^+4.81k^, with t in seconds and r in meters. At t=4.26 s, we have to find,(a) the x-component,(b) the y-component,(c) the magnitude, and(d) the angle relative to the positive direction of the x-axis, of the electron's velocity v (give the angle in the range (−180∘,180∘]) ?
The position vector of the electron is given as r=2.10ti^−4.99t²j^+4.81k^We can find the velocity by differentiating the position vector with respect to time.taking the derivative of r with respect to time,
we get v =[tex]2.10i^ - 9.98tj^ + 0k^[/tex] Velocity vector components arev_x = 2.10i^v_y = -9.98tj^v_z = 0
The magnitude of the velocity vector is given by,
|v| = √v_x² + v_y² + v_z²|v|
= √(2.10)² + (-9.98 × 4.26)² + 0|v|
= 24.06 m/s
The angle that the velocity vector makes with the positive x-axis is given by,
θ = tan⁻¹(v_y / v_x)
θ = tan⁻¹(-9.98 × 4.26 / 2.10)
θ = -85.56°
Therefore, the required values are as follows,
(a) The x-component is 2.10 m/s
(b) The y-component is -42.58 m/s
(c) The magnitude is 24.06 m/s
(d) The angle relative to the positive direction of the x-axis is -85.56°
Note: The direction of the angle is in the 4th quadrant.
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2.14 ×10
9
C charge has coordinates x=0,y=−2.00; a 3.09×10
9
C charge has coordinates x=3.00.y=0; and a −4.55×10
−9
.C charpe has coardinates x=3.00, y =4.00, where all distances are in cm. Determine magnitude and direction for the electric field at the origin and the instantianeous acceleration of a proton placed at the origin. (a) Determine the magrutude and direction for the electric field at the origin (measure the angle counterclockwise from the positive x-axis). magnitude direction (b) Determine the magnitude and direction for the instantanecus acceleration of a proton placed at the arigin (measure the angle. counterciockwise from the positive x-axis). magnitude direction
The magnitude and direction of the electric field at the origin, caused by the given charges, can be determined using the principle of superposition. The instantaneous acceleration of a proton placed at the origin can also be calculated based on the electric field. The answer to part (a) will provide the magnitude and direction of the electric field, while part (b) will provide the magnitude and direction of the proton's acceleration.
To determine the magnitude and direction of the electric field at the origin, we need to calculate the individual electric fields generated by each charge and then sum them up using vector addition. The electric field due to a point charge is given by the equation E = kq/r^2, where k is the electrostatic constant (8.99 × 10^9 N m^2/C^2), q is the charge, and r is the distance from the charge to the point of interest.
For the first charge (2.14 × 10^9 C) at coordinates (0, -2.00 cm), the distance from the origin is r1 = 2.00 cm. Using the equation above, we can calculate the electric field magnitude and direction. Similarly, for the second charge (3.09 × 10^9 C) at coordinates (3.00 cm, 0), the distance from the origin is r2 = 3.00 cm. Again, we can calculate the electric field magnitude and direction for this charge. Lastly, for the third charge (-4.55 × 10^(-9) C) at coordinates (3.00 cm, 4.00 cm), the distance from the origin is r3 = 5.00 cm. The electric field magnitude and direction can be determined for this charge as well.
To find the net electric field at the origin, we add up the electric field vectors from each charge using vector addition. The resulting vector will have a magnitude and direction that represents the net electric field at the origin.
For the instantaneous acceleration of a proton placed at the origin, we can use the equation F = qE, where F is the force experienced by the proton, q is the charge of the proton (1.60 × 10^(-19) C), and E is the electric field at the origin. Since force equals mass times acceleration (F = ma), we can rearrange the equation to find the acceleration (a = F/m), where m is the mass of the proton (1.67 × 10^(-27) kg).
Once the acceleration is determined, we can calculate the magnitude and direction of the proton's acceleration vector using the values obtained. The direction will be the same as the direction of the electric field at the origin.
In conclusion, by calculating the electric fields from the given charges and summing them up, we can determine the magnitude and direction of the electric field at the origin. Using this electric field, we can then find the instantaneous acceleration of a proton placed at the origin. The acceleration will have both magnitude and direction, indicating how the proton will move under the influence of the electric field.
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Translate Algebraic Expressions
5) Take away 7 from 4 times x
6) Add 6 to 3 times p
7) Subtract one-third from 9 times s
8) One-fifth of r is subtracted from 8
9) 6 times the sum of 8 and y
10) 3 is subtracted from five-sixths of c
