Answers
The measure of the angle μ∠ABD subtended by the arc AD at the circumference is equal to 27°
What is angle subtended by an arcThe angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.
arc AD = 2(μ∠ABD)
Also arc AD = 54°
2(μ∠ABD) = 54°
μ∠ABD = 54°/2 {divide through by 2}
μ∠ABD = 27°
Therefore, the measure of the angle μ∠ABD subtended by the arc AD at the circumference is equal to 27°
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Related Questions
the Dimensions of this figure are changed so that the new surface area is Exactly 1/4 What it was Originally. What is the New Surface Area? enter your Answer as a Decimal in the box.
Answers
The new surface area of the figure is given as follows:
S = 164.61 ft².
What is the surface area of a rectangular prism?The surface area of a rectangular prism of height h, width w and length l is given by:
S = 2(hw + lw + lh).
This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.
The prism in this problem is composed as follows:
6.5 ft, 9.8 - 4.3 = 5.5 ft and 10.6 ft.4.3 ft, 10.6 ft and 8.1 ft.Hence the surface area of the original prism is given as follows:
S = 2 x (6.5 x 5.5 + 5.5 x 10.6 + 6.5 x 10.6) + 2 x (4.3 x 10.6 + 4.3 x 8.1 + 10.6 x 8.1)
S = 658.44 ft².
The new surface area is one fourth of the original surface area, hence it is given as follows:
S = 0.25 x 658.44
S = 164.61 ft².
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For the surface with parametric equations r(s,t st, s + t, s-t) , find the equation of the tangent plane at (2, 3, 1). = Find the surface area under the restriction s4 + t2
Answers
The surface with parametric equation of the tangent plane to the surface at the point (2, 3, 1) is x - y - z = -2
To find the equation of the tangent plane to the surface described by the parametric equations r(s ,t) = (s, t+ s, s-t) at the point (2, 3, 1), we need to determine the partial derivatives of the position vector r(s, t) with respect to both s and t.
Let's calculate these derivatives:
∂r/∂s = (∂x/∂s, ∂y/∂s, ∂z/∂s)
= (1, 1, 1)
∂r/∂t = (∂x/∂t, ∂y/∂t, ∂z/∂t)
= (0, 1, -1)
Now, we can use the partial derivatives to find the normal vector to the tangent plane at the point (2, 3, 1). The normal vector is given by the cross product of the partial derivative vectors:
n = ∂r/∂s × ∂r/∂t
= (1, 1, 1) × (0, 1, -1)
Performing the cross product:
n = (1 * 1 - 1 * 0, 1 * (-1) - 1 * 0, 1 * 0 - 1 * 1)
= (1, -1, -1)
Since the normal vector is (1, -1, -1), we can use this vector as the coefficients of the equation of the tangent plane. The equation of a plane can be written as A x + By + C z = D, where (A, B, C) is the normal vector and (x, y, z) is a point on the plane.
Using the point (2, 3, 1) on the surface and the normal vector (1, -1, -1), the equation of the tangent plane becomes:
1 * x + (-1) * y + (-1) * z = D
x - y - z = D
To find the value of D, substitute the coordinates (2, 3, 1) into the equation:
2 - 3 - 1 = D
D = -2
Therefore, the equation of the tangent plane to the surface at the point (2, 3, 1) is:
x - y - z = -2.
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education and public health professionals are interested in how many adolescents between 15 and 18 years of age in a youth detention facility have a reading disability. a comprehensive screening program found that 38 adolescents were found to having a reading disability and 369 adolescents did not have a reading disability. the u.s. department of education estimates that about 9% of adolescents in this age group have a reading disability. conduct a hypothesis test to determine if there is significant evidence to suggest that the population of adolescents in a youth detention facility has a reading disability prevalence that is different than 9% (use a 0.05 significance level). what is the z-test value (test statistic for the hypothesis test)?
Answers
To determine if there is significant evidence to suggest that the population of adolescents in a youth detention facility has a reading disability prevalence that is different from 9%, we need to conduct a hypothesis test. Let p be the proportion of adolescents in the population who have a reading disability. Our null hypothesis is that p=0.09, and the alternative hypothesis is that p is not equal to 0.09. We will use a significance level of 0.05.
Using the sample data, we can calculate the sample proportion, p-hat, which is 38/407=0.0933. To calculate the z-test value, we first need to calculate the standard error, which is sqrt(0.09*(1-0.09)/407)=0.0191. The z-test value is (0.0933-0.09)/0.0191=1.57. This z-test value can be compared to the critical value of the standard normal distribution at a significance level of 0.05/2=0.025. The critical value is 1.96. Since the calculated z-test value of 1.57 is less than the critical value of 1.96, we fail to reject the null hypothesis. Therefore, there is not significant evidence to suggest that the population of adolescents in a youth detention facility has a reading disability prevalence that is different than 9%.
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I don’t understand this
Answers
4+(8+6)
4+14=18
The formula for associative property is (a+b)+c=a+(b+c)
An example would be:
(8+4)+2=8+(4+2)
Remembering this formula should be able to help you with associative property
Hope this helps
Basically
Associative property shows the association between three or more numbersIt shows that if we are adding or multiplying three or more numbers, the answer will not be changed in whatever order we multiply or add the three numbersFor example in addition:3+(5+2)= 10
3+7= 10
10=10
will give the same answer as
(3+5)+2=10
8+2=10
10=10
Hence the order of adding didn't affect our answer
Elena Wallace invested $150,000 in a project that pays her an even amount per year for 10 years. The payback period is 6 years. What are Elena's yearly cash inflows from the project? a. $150,000 b. $15,000 c. $25,000 d. $90,000 e. Cannot be determined from this information
Answers
Elena's yearly cash inflows from the project after the payback period is $15,000. A correct answer is an option (b).
The payback period is the time it takes for the project's cash inflows to equal the initial investment. In this case, the payback period is 6 years, meaning that after 6 years, Elena will have received enough cash inflows to recover her initial investment of $150,000.
Since the project pays Elena an even amount per year for 10 years, and the payback period is 6 years, she will receive cash inflows for an additional 4 years after the payback period. Therefore, to calculate Elena's yearly cash inflows, we divide the remaining cash inflows by the number of years remaining:
Remaining cash inflows = $150,000 (initial investment) - cash inflows received during the payback period
= $150,000 - ($15,000 x 6)
= $60,000
Yearly cash inflows = Remaining cash inflows/number of years remaining
= $60,000 / 4
= $15,000
Hence, B is the correct option.
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Which measure of center and measure of variability best describe the data set? Explain
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When two data sets are both symmetric, then the appropriate measure of center to describe them would be the mean.
How to explain the informationThe mean, also known as the average, is calculated by adding up all the values in the data set and dividing by the total number of values. It represents the "center" of the data because it balances out the values on both sides of the distribution.
The mean is a good measure of center for symmetric data sets because it captures the balance of the distribution and provides a single value that summarizes the data.
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Which measure of center should you use to describe two data sets that are both symmetric?
The ending inventory will form part of the items that were purchased in the period of rising prices. The cost of goods sold will be lower as the sales are not made from the current purchases. Hence, FIFO methof will produce the lowest amount of cost of goods sold in the period of rising prices.
Answers
The statement you provided is correct. In a period of rising prices, the cost of goods sold (COGS) will be lower if the items sold were purchased at a lower cost in a previous period. The ending inventory, on the other hand, will represent items purchased at a higher cost in the current period.
This is where the choice of inventory costing method comes into play. The FIFO (first in, first out) method assumes that the items sold are those that were purchased first, leaving the most recently purchased items in ending inventory. As a result, the COGS will reflect the lower cost of the earlier purchased items, leading to a lower COGS overall. Therefore, in a period of rising prices, the FIFO method will produce the lowest amount of COGS.
However, it is important to note that the choice of inventory costing method can also affect the valuation of ending inventory and ultimately impact the financial statements of a company.
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Use the definition of Taylor series to find the first three nonzero terms of the Taylor series (centered at c) for the function f. f(x) = 6 tan x, c = 5pi
Answers
The first three nonzero terms of the Taylor series are:
f(x) = 6(x-5π) + 0(x-5π)² + ... = 6x - 30π
What is the Taylor series?
A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The series provides a way to approximate the function in the neighborhood of that point.
We start by finding the nth derivative of f(x) at x = 5π for any positive integer n:
f(x) = 6 tan x
f'(x) = 6 sec² x
f''(x) = 12 sec² x tan x
f'''(x) = 12 sec⁴x + 24 sec² x tan² x
We can see a pattern emerging in the derivatives, so we can guess that the nth derivative is:
f^(n)(x) = P(n) secⁿx + Q(n) sec⁽ⁿ⁻²⁾x tan² x
where P(n) and Q(n) are polynomials in n.
Now, we can use the definition of the Taylor series:
f(x) = Σ0,∞(x-c)ⁿ
to find the first three nonzero terms of the Taylor series for f(x) centered at c = 5π.
Plugging in the nth derivative at x = 5π:
fⁿ(5π) = P(n) secⁿ 5π + Q(n) sec⁽ⁿ⁻²⁾ 5π tan² 5π
We can simplify this using the fact that sec(5π) = -1 and tan(5π) = 0:
fⁿ(5π) = (-1)ⁿ P(n) + Q(n) (-1)⁽ⁿ⁻¹⁾
Now, we can write out the first few terms of the Taylor series:
f(x) = f(5π) + f'(5π)(x-5π) + (f''(5π)/2!)(x-5π)² + ...
f(5π) = 6 tan(5π) = 0
f'(5π) = 6 sec²(5π) = 6
f''(5π) = 12 sec²(5π) tan(5π) = 0
hence, the first three nonzero terms of the Taylor series are:
f(x) = 6(x-5π) + 0(x-5π)² + ... = 6x - 30π
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use the table of values and produce a graph for the data. use the number of tables along the x-axis and the number of guests along the y-axis. plot each pair of values
Answers
By using the table of values, a graph of the number of tables along the x-axis and the number of guests along the y-axis is shown in the image below.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.Based on the sitting arrangement, we have:
x y_____
1 table 6 guests.
2 table 10 guests.
3 table 14 guests.
Next, we would determine the slope;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (10 - 6)/(2 - 1)
Slope (m) = 4/1
Slope (m) = 4
At data point (1, 6) and a slope of 4, a linear equation for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 6 = 4(x - 1)
y = 4x - 4 + 6
y = 4x + 2
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Unit 5 progress check: mcq part a ap calculus ab Let f be the function given by f(x)=5cos2(x2)+ln(x+1)−3. The derivative of f is given by f′(x)=−5cos(x2)sin(x2)+1x+1. What value of c satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4] ?
Answers
By the Mean Value Theorem, there exists a value c in the interval [1,4] such that f'(c) is equal to the average rate of change of f on the interval [1,4], which is (f(4) - f(1))/(4-1).
We can start by computing f(4) and f(1):
f(4) = 5cos(2(4^2)) + ln(4+1) - 3 = -0.841 + 1.609 - 3 = -1.232
f(1) = 5cos(2(1^2)) + ln(1+1) - 3 = 2.531 - 0.693 - 3 = -1.162
Then, we can compute the average rate of change:
(f(4) - f(1))/(4-1) = (-1.232 - (-1.162))/3 = -0.023
To satisfy the conclusion of the Mean Value Theorem, we need to find a value c in the interval [1,4] such that f'(c) = -0.023. From the given expression for f'(x), we can see that there is no value of c that satisfies this equation, since f'(x) can never be negative. Therefore, there is no value of c that satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4].
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Which equation has the same unknown value as
323 ÷ 17?
Answers
Answer:
B. 17 * unknown number = 323
Step-by-step explanation:
Let's call the unknown number n. Thus 323 / 17 = n
Since we know that 323 / 17 = n, we get 323 by multiplying 17 and n.
Thus, our answer is B.
Other example: Let's use 20 / 4 as an example. We know that 20 / 4 = 5. Thus, 4 * 5 = 20, where 5 is the answer to division problem but one of the products in the multiplication problem.
Gerald earns $___ each time he mows a lawn. He mows his neighbor's lawn each week. He also earned an additional $___ for trimming trees one week.
Part A: Rewrite the description by filling in the blanks with values of your choice to show the amount of money he could earn in any number of weeks, w. Make sure that the values you choose make sense for this situation. (1 point)
Part B: Write an algebraic expression from your written description used in Part A. Let w stand for the number of weeks. (3 points)
Answers
The algebraic expression representing the description given is 5w + 2w
We can rewrite the description as Gerald earns $5 each time he mows a lawn
Rewriting the description:Let amount earned for lawn mowing = $5Let amount earned for trimming = $2
We can then rewrite the description as Gerald earns $5 each time he mows a lawn. He mows his neighbor's lawn each week.
He also earned an additional $2 for trimming trees one week.
An algebraic expression for the description
Number of weeks = w
Amount earned for any given number of weeks is ;
Amount earned= 5w + 2w = 7w
Hence, the algebraic expression for the description is 5w + 2w
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If f(x) = x² + 5x - 7, find the following.
2. f(-1)
Answers
Answer:
keeping the value of x as -1
then we have
[tex]f( - 1) = {1}^{2} + 5 \times 1 - 7[/tex]
=1 +5-7
= -1
what angle (in degrees) corresponds to 6 rotations around the unit circle?
Answers
Answer:
Six rotations around the unit circle correspond to 2160 °.
Step-by-step explanation:
We know that One rotation around a circle is equal to 360 degrees :
i.e. 1 rotation = 360 ° ........(i)
Hence, 6 rotations = ( 6 × 360 ° )
So for 6 rotations, we have 2160 °.
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please help asap i need to get my grade up
Answers
Answer:
sin I = 3/5
Step-by-step explanation:
sin I = perpendicular/hypotenuse
= 18/30
= 9/15
= 3/5
There are 250 seventh graders going on a field trip to the zoo and 9 will be selected to feed the animals. Which method ensures a random sample is selected to feed the animals?
Answers
The best way or method that can ensure a random sample is selected to feed the animals would be simple random sampling method.
What is simple random sampling method?The simple random sampling is a sampling method when all the members of s population data set is given an equal opportunity to be selected for a research work.
This type of sampling method is usually carried out when there are large group of people to choose few from with respect to a research to be conducted.
Therefore, the 9 individuals that would be selected for the feeding of animals will require the use of simple random sampling.
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Find the 19th term of a geometric sequence where the
first term is -6 and the common ratio is -2.
Answers
Answer:
-1572864
Step-by-step explanation:
You want the 19th term of the geometric sequence with first term -6 and common ratio -2.
N-th termThe n-th term of a geometric sequence is ...
an = a1·r^(n-1)
where a1 is the first term, and r is the common ratio.
Using the given values of a1 and r, the 19th term is ...
a19 = (-6)·(-2)^(19-1) = -1572864
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Find the derivative for y = (2x - 1)^3(4x + 3)^-3
Answers
Answer:
Differentiate using the Product Rule,
[tex]\frac{d}{dx}[/tex][f(x)g(x)]=f(x)[tex]\frac{d}{dx}[/tex][g(x)]+g(x)[tex]\frac{d}{dx}[/tex][f(x)]
[tex]\frac{30(2x-1)^2}{(4x+3)^4}[/tex]
Step-by-step explanation:
What is the slope?
Simplify the answer and write it as a proper fraction, improper fraction, or integer.
Answers
The y-intercept and the slope of the linear equation will be 2 and 50, respectively.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
The slope of the line is given as,
m = (y₂ - y₁) / (x₂ - x₁)
From the graph, the y-intercept is 50. Then the slope is calculated as,
m = (70 - 50) / (10 - 0)
m = 20 / 10
m = 2
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Please help me I don't understand this with steps please
Answers
The surface area of the rectangular prism is 1236 in².
We have,
The surface area of the rectangular prism.
= lower surface + top surface + back surface + front surface
+ 2 x side surface
Each surface is in the form of a rectangle.
So,
= 14 x 8 + 14 x 8 + 23 x 8 + 23 x 8 + 2 x (23 x 14)
= 112 + 112 + 184 + 184 + 2 x 322
= 112 + 112 + 184 + 184 + 644
= 1236 in²
Thus,
The surface area of the rectangular prism is 1236 in².
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Please help I need the answer right mow
Answers
System A has 2 real solutions. System B has do not have more than 2 real solutions. System C has do not have more than 2 real solutions.
What is a real solution?A real solution in algebra is described simply as a solution to an equation that is a real number.
For system A The equation x² + y² = 17 represents a circle with center (0,0) and radius √17 and the x-axis intersects two times same with the y-axis.
Therefore, the system has 2 two real solutions. for System B:The equation y = x² - 7x + 10 represents a parabola with vertex at (3.5, -1.25).
System C:The equation y = -2x² + 9 shows a parabola facing down with vertex at (0,9/2).
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Use your understanding of angle relationships to set up and solve an equation to find the missing angle measures. Pls help !
Answers
The property to be used is vertical angle theorem and the value of x is 20/3.
Given is a figure in which two lines are intersecting at a point, making two angles,
The angles are = 3x and 20°,
We need to determine the value of x and the property involved.
So, according to figure we can say, the property involved is vertical angle theorem.
Therefore,
3x = 20
x = 20/3
Hence the property to be used is vertical angle theorem and the value of x is 20/3.
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find integral from (1)^e(1/z 1/[z^2])dz (e = 2.71 dots).
Answers
The integral from (1) to e of [(1/z) - (1/z^2)] dz equals ln(e) - ln(1) - (1 - 1/e) = 1 - 1/e.
To solve the integral, we can use the power rule of integration. First, we split the integral into two parts: ∫(1 to e) 1/z dz - ∫(1 to e) 1/z^2 dz.
For the first part, we integrate 1/z with respect to z, which gives us ln|z|. Evaluating this from 1 to e, we get ln|e| - ln|1| = ln(e) - ln(1) = 1.
For the second part, we integrate 1/z^2 with respect to z, which gives us -1/z. Evaluating this from 1 to e, we get -1/e + 1.
Finally, we subtract the result of the second part from the result of the first part, giving us 1 - 1/e.
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At even time instants, a robot moves either +4 cm or -A cm in the x-direction according to the outcome of a coin flip; at odd time instants, a robot moves similarly according to another coin flip in the y-direction. Assuming that the robot begins at the origin, let X and Y be the coordinates of the location of the robot after 2n time instants. (a) Describe the underlying space 12 of this random experiment and show the mapping from 1 to 1xy, the range of the pair (X,Y). (b) Find the marginal pmf of the coordinates X and Y. (c) Find the probability that the robot is within distance V2 of the origin after 2n time instants.
Answers
(a) The underlying space Ω consists of all possible sequences of coin flips, mapping to the range of the pair (X,Y) representing the coordinates of the robot after 2n time instants. (b) The marginal pmf of X is P(X = -4) = P(Tails) and P(X = 4) = P(Heads), while the marginal pmf of Y is P(Y = -A) = P(Tails) and P(Y = A) = P(Heads). (c) The probability that the robot is within distance V/2 of the origin after 2n time instants depends on the specific probabilities associated with the coin flips and the value of A.
(a) The underlying sample space Ω of this random experiment consists of all possible sequences of coin flips. Each coin flip can result in either a "heads" or "tails" outcome, corresponding to +4 cm or -A cm movement in the x-direction. The sequences of coin flips determine the movements of the robot at even and odd time instants.
The mapping from the sample space Ω to the range of the pair (X,Y) can be described as follows:
1 -> x: -4 cm, y: 0
2 -> x: 0, y: -A cm
3 -> x: 0, y: 0
4 -> x: 4 cm, y: 0
5 -> x: 0, y: A cm
6 -> x: 0, y: 0
...
Each coin flip outcome corresponds to a particular movement in either the x or y direction, and the resulting coordinates (X,Y) are determined by the cumulative movements after 2n time instants.
(b) To find the marginal pmf of the coordinates X and Y, we need to calculate the probabilities associated with each possible value of X and Y.
Since at even time instants the robot moves either +4 cm or -A cm in the x-direction, the pmf of X can be described as:
P(X = -4) = P(Tails)
P(X = 4) = P(Heads)
Similarly, at odd time instants, the robot moves either +4 cm or -A cm in the y-direction, resulting in the pmf of Y as:
P(Y = -A) = P(Tails)
P(Y = A) = P(Heads)
(c) To find the probability that the robot is within distance V/2 of the origin after 2n time instants, we need to consider the possible combinations of movements that result in the robot being within this distance.
For example, if V = 8 cm, the robot can be within distance V/2 of the origin if it has moved +4 cm or -4 cm in either the x or y direction.
To calculate the probability, we need to sum the probabilities of the corresponding movements in the x and y directions:
P(|X| ≤ V/2, |Y| ≤ V/2) = P(X = -4) * P(Y = 0) + P(X = 4) * P(Y = 0) + P(X = 0) * P(Y = -A) + P(X = 0) * P(Y = A)
This calculation will depend on the specific probabilities associated with the coin flips and the value of A.
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compnay a charges $82 and allows unlimited mileage. company b has an intial fee of $55 and charges an additional $0.60 for every mile driven. for what mileage will company a charge less than company b
Answers
For distances of 45 miles or less, Company A is cheaper, while for distances greater than 45 miles, Company B is the cheaper option.
To determine at what mileage Company A charges less than Company B, we can set up an equation and solve for the variable, which in this case will represent the number of miles driven. Let x be the number of miles driven, and let C(x) represent the cost of renting a car from Company B after driving x miles.
We know that Company A charges a flat fee of $82 for unlimited mileage, so we can represent the cost of renting from Company A as a constant function C(x) = 82. For Company B, the cost function is given by:
C(x) = 55 + 0.60x
We want to find the value of x for which Company A charges less than Company B. In other words, we want to find the point at which the two cost functions intersect. To do this, we can set the two functions equal to each other and solve for x:
82 = 55 + 0.60x
27 = 0.60x
x = 45
Therefore, when the number of miles driven is 45 or less, Company A charges less than Company B. For any mileage greater than 45, it is cheaper to rent from Company B.
In summary, Company A charges a flat rate of $82 for unlimited mileage, while Company B charges an initial fee of $55 and an additional $0.60 for every mile driven. To find the point at which Company A charges less than Company B, we set the two cost functions equal to each other and solve for the number of miles driven. The result is 45 miles, meaning that for any distance of 45 miles or less, it is cheaper to rent from Company A, while for any distance greater than 45 miles, Company B is the cheaper option.
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3. What transformations on the graph f(x) = loga x result in the graph of g(x) = -logs (x + 5)?
Answers
The transformations on the graph f(x) = loga x result in the graph of g(x) = -logs (x + 5) is found when we translate 5 units to the left then reflect across the x-axis.
What is graph transformations?Graph transformation is described as the process by which an existing graph, or graphed equation, is modified to produce a variation of the proceeding graph.
Some available graph transformations includes:
TranslationDilation ReflectionSo if we translate 5 units to the left then reflect across the x-axis on the graph f(x) = log x, the result is in the graph of g(x) = -logs (x + 5)
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need to find m angel S
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60 is your answer
The measure of the angle S is 60 degrees
How to determine the angleTo determine the angle, we need to know the six different trigonometric identities in mathematics;
These trigonometric identities are;
sinecosinetangentcotangentsecantcosecantWe also have that these identities have their ratios, we have that;
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
From the information given, we have that;
Adjacent = 2√3
Hypotenuse= 4√3
Then,
cos S = 2√3/4√3
Divide the values
cos S = 0. 5
find the inverse
S = 60 degrees
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helpppp show work pls
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Step-by-step explanation:
hope this helps if this wasn't what you looking for sorry
find the missing coordinates such that the three vectors form an orthonormal basis for r3 : [ -0.8 ] -0.6 0 , [ ] -1 , [ ] -0.8 .
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The missing coordinates of the three vectors form which makes them an orthonormal basis for R³ are as follow,
v₁ = [-0.8, -0.6, 0]
v₂ = [-0.6, -1, 0.45]
v₃ =[-0.27, -0.36, -0.8].
To form an orthonormal basis for R³, the three vectors must be orthogonal that is perpendicular to each other.
And have unit length norm equal to 1.
Two of the vectors, find the missing coordinates to satisfy these conditions.
Let us consider the two given vectors,
v₁ = [-0.8, -0.6, 0]
v₂ = [?, -1, ?]
To find the missing coordinates of v₂,
Find a vector that is orthogonal to v₁.
One way to do this is by taking the cross product of v₁ and v₂, which will give us a vector orthogonal to both.
Cross product formula: v₁ × v₂ = [a₁b₂ - a₂b₁, a₂b₀ - a₀b₂, a₀b₁ - a₁b₀]
Using the cross product formula, find the missing coordinates of v₂,
v₂ = [?, -1, ?] = v₁ × [?, -1, ?]
Let us calculate the cross product,
v₂
= [?, -1, ?]
= [-0.8 × ?, -0.6 × (-1) - 0 × ?, 0 × ? - (-0.6 × ?)]
To satisfy the orthogonality condition, the dot product of v₁ and v₂ must be zero,
v₁ · v₂ = -0.8 × ? + (-0.6) × (-1) + 0 × ?
⇒ -0.8 × ? + (-0.6) × (-1) + 0 × ? = 0
Simplifying the equation,
⇒-0.8 × ? + 0.6 + 0 = 0
⇒ -0.8 × ? = -0.6
Dividing both sides by -0.8,
⇒ ? = -0.6 / -0.8
⇒ ? = 0.75
Now substitute this value back into the cross product equation to find the missing coordinates of v₂,
v₂ = [-0.8 × 0.75, -1, 0.6 × 0.75]
= [-0.6, -1, 0.45]
The missing coordinates for the vector v₂ are [-0.6, -1, 0.45].
To find the missing coordinates for the third vector,
Use the same process.
Let us consider the two given vectors,
v₁ = [-0.8, -0.6, 0]
v₂ = [-0.6, -1, 0.45]
v₃ = [?, ?, ?]
Again, find a vector that is orthogonal to both v₁ and v₂.
Use the cross product to determine the missing coordinates,
v₃ = [?, ?, ?]
= v₁ × v₂
Calculating the cross product,
⇒ v₃ = [?, ?, ?]
= [-0.6 × 0.45 - 0 × (-1), 0 × (-0.6) - (-0.8 × 0.45), (-0.8) × (-1) - (-0.6) × 0]
Simplifying the equation,
⇒v₃ = [?, ?, ?]
= [-0.27, -0.36, -0.8]
The missing coordinates for the vector v₃ are [-0.27, -0.36, -0.8].
Therefore, the missing coordinates that would make the three vectors form an orthonormal basis for R³ are,
v₁ = [-0.8, -0.6, 0]
v₂ = [-0.6, -1, 0.45]
v₃ =[-0.27, -0.36, -0.8].
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the proportion of college football players who have had at least one concussion is estimated to be 34% in the united states. we wanted to know if football players at our university were less likely to have suffered a concussion, so we surveyed a random sample of 100 past and present football players at our university. is this survey valid or not valid for testing the hypothesis that the proportion of college football players at our university with at least one concussion is less than the national average?
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All of the criteria's are fulfilled the survey is valid.
We have the information from the question:
The proportion of college football players who have had at least one concussion is estimated to be 34% in the united states.
Then, 34% = 0.34
The sample size of the data is = 100
p: the ‘proportion’ of ‘college’
The required conditions for testing the hypothesis of population proportion are,
(i) The population is larger than the sample
(ii) np > 10
=> 100 × 0.34
=34
(iii) n(1-p) > 10
100 × (1 - 0.34)
=> 100 × 0.66
=66
iv)The ‘sample’ is drawn randomly from the population.
Since all of the above criteria's are fulfilled the survey is valid.
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