The matrix exponential M(t) = exp(t*A) are: M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)
To find the matrix exponential M(t) = exp(t*A), we first need to find the eigenvectors of A corresponding to the eigenvalues X1 = 1 and X2 = 2.
For X1 = 1, we solve the equation (A - I)*v = 0, where I is the identity matrix:
(A - I)*v = (1 1; 2 2 - 1)*v = 0
RREF([A - I, zeros(2,1)])
ans =
0 0 -1
0 0 0
So we have the equation -v2 = 0, which means v2 can be any non-zero value. Let's choose v2 = 1, then v1 = -1/2. So the eigenvector corresponding to X1 is v1 = (-1/2; 1).
For X2 = 2, we solve the equation (A - 2*I)*v = 0:
(A - 2*I)*v = (-1 1; 2 -2)*v = 0
RREF([A - 2*I, zeros(2,1)])
ans =
0 0 -1
0 0 0
So we have the equation -v2 = 0, which means v2 can be any non-zero value. Let's choose v2 = 1, then v1 = 1/2. So the eigenvector corresponding to X2 is v2 = (1/2; 1).
Now we can construct the matrix exponential M(t) = exp(t*A) using the formula:
M(t) = c1*exp(X1*t)*v1*v1' + c2*exp(X2*t)*v2*v2'
where c1 and c2 are constants determined by the initial conditions. Since we don't have any initial conditions given, we can choose c1 = 1 and c2 = 0 for simplicity.
So we have:
M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)
So the matrix exponential M(t) is:
M(t) = ( (1/4)*exp(t) + (1/2)*exp(2*t) (1/2)*(exp(2*t) - exp(t));
(1/2)*(exp(2*t) + 1) exp(t) + exp(2*t) )
To find the matrix exponential M(t) = exp(tA) given that the eigenvalues of matrix A are λ1 = 1 and λ2 = 2, we first need to find the eigenvectors corresponding to each eigenvalue, and then form the matrix P of eigenvectors and the diagonal matrix D of eigenvalues. Finally, we can compute M(t) using the formula:
M(t) = P * exp(tD) * P^(-1)
After finding the eigenvectors and forming the matrices P and D, compute exp(tD) by taking the exponentiation of each diagonal element:
exp(tD) = | exp(tλ1) 0 |
| 0 exp(tλ2) |
Now, compute M(t) by multiplying P, exp(tD), and the inverse of P. The resulting matrix M(t) will have the following components:
M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)
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when a third variable is included in the analysis that is studying the relationship between an independent variable and a dependent variable, and this third variable changes the relationship between the independent variable and the dependent variable in an important way, this third variable is called a(n): a. moderator variable b. outlier variable c. spurious variable d. contingency variabl
A moderator variable is a variable that affects the strength or direction of the relationship between an independent variable and a dependent variable. In other words, it influences the degree to which the independent variable impacts the dependent variable.
When a third variable is included in the analysis, it is important to identify its role in the relationship between the independent and dependent variables. If the third variable changes the relationship between the two variables in an important way, then it is likely acting as a moderator variable. This means that the relationship between the independent and dependent variables is not as straightforward as originally thought, and that the third variable must be considered when analyzing the relationship.
For example, imagine a study that examines the relationship between exercise and weight loss. The independent variable is exercise, the dependent variable is weight loss, and a third variable could be age. If age is found to moderate the relationship between exercise and weight loss (i.e., older individuals may not experience the same weight loss benefits from exercise as younger individuals), then age is considered a moderator variable.
In summary, including a third variable in the analysis can reveal important information about the relationship between the independent and dependent variables. A moderator variable specifically changes the strength or direction of this relationship and must be carefully considered during analysis.
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1. a 24 centimeter chord is 32 centimeters from the center of a circle. find the length of the radius.
The length of the radius of the circle is [tex]4\sqrt{73}[/tex].
Let, AB is the chord with a length of 24 centimeters. Line OD is the distance between the chord and the center of the circle. Line r is the radius.
By the chord bisector theorem, line OD bisects the chord AB and it is perpendicular to AB.
Thus, the length of segment AD would be half of the length of AB. i.e,
AD = 12.
As triangle AOD is a right triangle, use Pythagoras theorem in triangle AOD to find the radius of the circle.
[tex]OA^{2} =AD^{2} +OD^{2} \\r^{2} =12^{2} +32^{2} \\r^{2} =144+1024[/tex]
Further simplify,
[tex]r^{2} =\sqrt{1168}\\ r=4\sqrt{ 73[/tex]
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Select the correct answer. Which equation could be solved using this application of the quadratic formula? A. -2x2 − 8 = 10x − 3 B. 3x2 − 8x − 10 = 4 C. 3x2 + 8x − 10 = -8 D. -2x2 + 8x − 3 = 4 Reset Next
Answer:
B
Step-by-step explanation:
The quadratic formula is used to solve quadratic equations in the form ax^2 + bx + c = 0.
Looking at the given options, we can see that option B can be written in this form as 3x^2 - 8x - 14 = 0. Therefore, the equation that could be solved using the quadratic formula is option B.
Question 6 of 15
The volume of a cylinder is 6377 cm³. If the radius is 3 cm, what is the height
of the cylinder?
3 cm
Answer:
The answer is 7cm
Step-by-step explanation:
Each week you collect 20 cards. Your friend collects 12 cards each week. How many cards does your friend have if you have 240 cards?
If you have 240 cards and collect 20 cards per week, you have 96 cards after 8 weeks and your freind have 240 cards in 7.5 weeks.
First, we need to find the total number of cards collected per week by both you and your friend
Total cards collected per week = your cards + friend's cards
Total cards collected per week = 20 + 12
Total cards collected per week = 32
Now, we can find the number of weeks it would take for your friend to collect 240 cards
240 cards ÷ 32 cards per week = 7.5 weeks
Since we cannot have a fractional number of cards, we need to round up to the nearest whole number of weeks. Therefore, it would take your friend 8 weeks to collect 240 cards.
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Question 2 of 10
Which situation is most likely to have a constant rate of change?
OA. Number of flowers in a flower bed compared with the area planted
B. The total amount paid for gas compared with the number of
gallons purchased.
C. Distance a delivery truck travels compared with the number of
deliveries made
D. Points scored in a basketball game compared with the number of
quarters played
Answer: The situation that is most likely to have a constant rate of change is option B: "The total amount paid for gas compared with the number of gallons purchased."
This is because the price of gas per gallon is usually constant, so the rate of change of the total amount paid for gas should be constant with respect to the number of gallons purchased. In other words, if you plot the total amount paid for gas against the number of gallons purchased, you would expect a straight line with a constant slope.
In contrast, the number of flowers in a flower bed compared with the area planted (option A), the distance a delivery truck travels compared with the number of deliveries made (option C), and points scored in a basketball game compared with the number of quarters played (option D) are less likely to have a constant rate of change because they can be affected by various factors such as weather, traffic, player performance, and so on.
Answer:
B.
Step-by-step explanation:
plot the point A(-2, -3) B(-2,2) C (3,2) and D (3,-3) on a number plane and join them together
a) what shape is formed
b) What is the length of AD
c) Find the perimeter ABCD
d) Now join points B and D. What is the are of BCD
The shape formed is rectangle. Length of AD is 5. Perimeter of ABCD is 20. Area of BCD is [tex]\sqrt{65}[/tex] .
a) The shape formed is a rectangle.
b) The length of AD can be found using the distance formula:
AD = [tex]\sqrt{(3-(-2))^{2}+(-3-(-3))^{2} }[/tex]
= [tex]\sqrt{5^{2}}[/tex]
= 5
Therefore, the length of AD is 5.
c) The perimeter of ABCD can be found by adding up the lengths of all four sides:
AB = [tex]\sqrt{(-2-(-2))^{2}+(2-(-3))^{2} }[/tex]
= [tex]\sqrt{5^{2}}[/tex]
= 5
BC = [tex]\sqrt{(3-(-2))^{2}+(2-2)^{2} }[/tex]
= [tex]\sqrt{5^{2}}[/tex]
= 5
CD = [tex]\sqrt{(3-3)^{2}+(-3-2)^{2} }[/tex]
= [tex]\sqrt{5^{2}}[/tex]
= 5
DA = [tex]\sqrt{(-2-3)^{2}+(-3-(-3))^{2} }[/tex]
= [tex]\sqrt{5^{2}}[/tex]
= 5
Perimeter = AB + BC + CD + DA
= 5 + 5 + 5 + 5
= 20
Therefore, the perimeter of ABCD is 20.
d) Now join points B and D to form line segment BD. The area of triangle BCD can be found using the formula for the area of a triangle:
Area of BCD = (1/2) * base * height
The base is BD, which has length:
BD = [tex]\sqrt{(3-(-2))^{2}+(-3-2)^{2} }[/tex]
= [tex]\sqrt{65}[/tex]
To find the height, we need to draw a perpendicular line from C to line BD:
The height is the length of the perpendicular line from C to line BD. Since C and D have the same x-coordinate, this perpendicular line will be vertical and have length 2 units (the difference between the y-coordinates of C and D).
Therefore, the height is 2.
Area of BCD = (1/2) * BD * height
= (1/2) * [tex]\sqrt{65}[/tex] * 2
= [tex]\sqrt{65}[/tex]
Therefore, the area of BCD is [tex]\sqrt{65}[/tex] square units.
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Dana buys a plant that is 4 inches tall. After one week the plant is 7 inches tall. After a second week the plant is 10 inches tall. At this rate, how tall will the plant be after the fifth week?
a
22 inches tall
b
3 inches tall
c
14 inches tall
d
19 inches tall
Answer:
We know that the plant grows by 3 inches each week (7 inches - 4 inches = 3 inches, and 10 inches - 7 inches = 3 inches). Therefore, after 5 weeks, the plant will be 4 inches + (3 inches × 5) = 19 inches tall.
Step-by-step explanation:
- The plant is 4 inches tall when Dana buys it.
- After one week, the plant grows by 3 inches to reach a height of 7 inches.
- After a second week, the plant grows by another 3 inches to reach a height of 10 inches.
- So, the plant grows by 3 inches each week.
- After 3 weeks, the plant will be 10 inches + 3 inches = 13 inches tall.
- After 4 weeks, the plant will be 13 inches + 3 inches = 16 inches tall.
- After 5 weeks, the plant will be 16 inches + 3 inches = 19 inches tall.
Therefore, the correct answer is d) 19 inches tall.
You draw two simple random samples from two distinct populations and calculate the following:= 23. 4, s1 = 4. 2, n1 = 25= 25. 3, s2 = 3. 9, n2 = 27The estimate of the degrees of freedom, k, equals n1 - 1, or 24, t* is 2. 064, and m, the margin of error, is 2. 325. Construct a 95% confidence interval for the difference between these two populations and draw a conclusion based on this confidence interval. Rnrm. Gif A. The confidence interval is (-4. 225,. 425); there's a difference between the two population means. Rnrm. Gif B. The confidence interval is (-6. 699, 2. 899); there's no difference between the two population means. Rnrm. Gif C. The confidence interval is (-3. 275, 1. 375); there's a difference between the two population means. Rnrm. Gif D. The confidence interval is (-4. 225,. 425); there's no difference between the two population means. Rnrm. Gif E. The confidence interval is (-6. 699, 2. 899); there's a difference between the two population means
The information given shows that E. The confidence interval is (-6. 699, 2. 899); there's a difference between the two population means
How to explain the confidence intervalCI = (x1 - x2) ± t* × m
Plugging in the given values, we get:
CI = (23.4 - 25.3) ± 2.064 × 2.325
= -1.9 ± 4.798
= (-6.698, 2.898)
Therefore, the 95% confidence interval for the difference between the two population means is (-6.698, 2.898).
Since this interval does not include zero, we can conclude that there is a statistically significant difference between the two population means at the 95% confidence level. The correct answer is option E.
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Which equation can be used to solve for
�
xx in the following diagram?
Choose 1 answer:
Choose 1 answer:
(Choice A)
150
−
10
�
=
90
150−10x=90150, minus, 10, x, equals, 90
A
150
−
10
�
=
90
150−10x=90150, minus, 10, x, equals, 90
(Choice B)
10
�
+
150
=
180
10x+150=18010, x, plus, 150, equals, 180
B
10
�
+
150
=
180
10x+150=18010, x, plus, 150, equals, 180
(Choice C)
10
�
=
150
10x=15010, x, equals, 150
C
10
�
=
150
10x=15010, x, equals, 150
(Choice D)
10
�
+
90
=
180
10x+90=18010, x, plus, 90, equals, 180
D
10
�
+
90
=
180
10x+90=180
An equation can be used to solve for x in the following diagram:
x + (4x - 85) = 90
The correct answer is an option (A)
From the attached figure we can observe that the right angle is divided into two angles i.e., angle x degree and angle (4x - 85) degrees
We know that the two angles are called complementary angles when the sum of their is equal to 90 degrees.
We can observe that angle x degree and angle (4x - 85) degrees are complementary angles.
This means that the sum of these angles must be 90 degrees.
x + (4x - 85) = 90
this is the required equation.
Thus, the correct answer is an option (A)
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Find the complete question below.
need help fast its due in 5 minutes write this number in standard form
Answer:
The answer to the question given is 785.639 .
Step-by-step explanation:
According to the rules of mathematics i.e. BODMAS
We'll first multiply,
7x100 = 700
8x10 = 80
5x1 = 5
6x[tex]\frac{1}{10}[/tex] = [tex]\frac{3}{5}[/tex] = 0.6
3x[tex]\frac{1}{100}[/tex] = 0.03
9x [tex]\frac{1}{1000}[/tex] = 0.009
So, after adding.
We get,
700 + 80 + 5 + 0.6 + 0.03 + 0.009 = 785.639
A reputable polling organization in a certain country surveyed 106,600 ​adults, and 18​% of those polled reported that they smoked. Complete parts a and b below.
b) Explain what this margin of error means. Select the correct choice below and fill in the answer box within your choice.
​(Round to four decimal places as​ needed.)
A.The probability that any given adult surveyed from the population smokes is ________________.
B.The probability that any given adult surveyed from the sample smokes is _____________.
C.We are 90​% confident that the observed proportion of adults that smoke is within _________of the sample proportion.
D.We are 90​% confident that the observed proportion of adults that smoke is within ________ of the population proportion
Both options C and D are correct in describing the meaning of the margin of error, but we cannot provide specific values for the margin of error without additional information.
To answer this question, first, we need to calculate the sample proportion of adults who smoke.
Calculate the sample proportion
Number of adults surveyed = 106,600
Percentage of adults who smoke = 18%
Sample proportion (p) = (Percentage of adults who smoke) / 100
p = 18% / 100 = 0.18
Now, let's address each option in part b:
A. The probability that any given adult surveyed from the population smokes is not the correct interpretation of the margin of error.
B. The probability that any given adult surveyed from the sample smokes is not the correct interpretation of the margin of error.
C. We are 90% confident that the observed proportion of adults that smoke is within the margin of error of the sample proportion.
To calculate the margin of error, we need more information, such as the standard deviation of the population and the desired confidence level.
Since we do not have this information, we cannot provide a specific value for the margin of error.
D. We are 90% confident that the observed proportion of adults that smoke is within the margin of error of the population proportion.
Similar to option C, we need more information to calculate the margin of error, so we cannot provide a specific value for the margin of error.
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An article reported on a school​ district's magnet school programs. Of the 1928 qualified​ applicants, 986 were​ accepted, 297 were​ waitlisted, and 645 were turned away for lack of space. Find the relative frequency for each decision made and write a sentence summarizing the results.
51.1% of the qualified applicants were accepted into the magnet school programs, 15.4% were waitlisted, and 33.5% were turned away due to a lack of space.
To find the relative frequency for each decision made by the school district's magnet school programs, we need to divide the number of applicants for each decision by the total number of qualified applicants.
Accepted applicants: 986 / 1928 = 0.511 or 51.1%
Waitlisted applicants: 297 / 1928 = 0.154 or 15.4%
Turned away applicants: 645 / 1928 = 0.335 or 33.5%
In summary, 51.1% of the qualified applicants were accepted into the magnet school programs, 15.4% were waitlisted, and 33.5% were turned away due to a lack of space.
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solve for z in the following equation $1 iz 1 iz where i 2 1 simplify your answer as much as possible
The solution for z is 0.
What is the equivalent expression?
Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.
Assuming the equation is:
1 + iz = 1 - iz
We can start by isolating the term with z on one side:
1 + iz = 1 - iz
2iz = 0
Divide both sides by 2i:
z = 0
Therefore, the solution for z is 0.
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Complete Question:
Solve for z in the following equation: 1-iz = -1 + iz (where i^2 = -1).
Simplify your answer as much as possible.
If Maira drives east from Atlanta to
Augusta in 2.5 hours. If her average
speed is 55 miles/hour, how far is
Augusta from Atlanta?
Answer:
137.5 miles
Step-by-step explanation:
To calculate the distance between Atlanta and Augusta, we can use the formula:
Distance = Speed x Time
We are given the speed and time, so we can substitute those values into the formula and solve for the distance.
Distance = 55 miles/hour x 2.5 hours
Distance = 137.5 miles
Therefore, Augusta is 137.5 miles away from Atlanta.
Answer: The answer is 147 miles per hour
Step-by-step explanation:
Find the height of the triangular pyramid when the volume is 318 square centimeters
The height of the triangular pyramid, given that the volume is 318 square centimeters 7.31 cm (option B)
How do i determine the height of the triangular pyramid?First, we shall obtain the base area of the triangular pyramid. Details below:
Base length (b) = 29 cmBase height (h) = 9 cmBase area (A) =?A = ½bh
A = ½ × 29 × 9
A = 130.5 cm²
Finally, we shall determine the height of the triangular pyramid. Details below:
Volume of triangular pyramid (V) = 318 cm³Base area of triangular pyramid (A) = 130.5 cm²Height of triangular pyramid (h) =?V = ⅓Ah
318 = ⅓ × 130.5 × h
318 = 43.5 × h
Divide both sides by 43.5
h = 318 / 43.5
h = 7.31 cm
Thus, the height of the triangular pyramid is 7.31 cm (option B)
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Complete question:
See attached photo
Use the following pattern and inductive reasoning to predict the answer to 9 x 7,654,321 - 1 .
Using the following pattern and inductive reasoning, predicted answer to 9 x 7,654,321 - 1 is 73,888,889.
The pattern:
When you subtract 1 from a number that ends with a sequence of n consecutive digits (all equal to d), the result is a number that ends with the same n digits followed by ([tex]10^{n}[/tex] - 1) -d.
For example:
If you subtract 1 from a number that ends with three 7's, the result is a number that ends with three 6's, i.e., (777-1=776).
If you subtract 1 from a number that ends with four 2's, the result is a number that ends with four 1's, i.e., (2222-1=2221).
Applying this pattern to 9 x 7,654,321 - 1:
The number ends with one 9, so n=1 and d=9.
Therefore, the result will end with one 8 (one less than 9), followed by ([tex]10^{1}[/tex] - 1) - 9 = 0, i.e., it will end with 8.
So, the predicted answer is 73,888,889.
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Solve the given differential equation by variation of parameters. Xy'' − 6y' = x6 y(x) = , x > 0
In this problem, we are given a differential equation involving the second derivative of a function y with respect to x. We are asked to solve the equation using the method of variation of parameters.
The differential equation we are given is:
xy'' - 6y' = [tex]x^6[/tex], x > 0
To solve this equation by variation of parameters, we first need to find the general solution of the corresponding homogeneous equation, which is:
xy'' - 6y' = 0
To do this, we assume that the solution has the form y = c[tex]x^m[/tex], where c is a constant and m is a power of x. Substituting this into the homogeneous equation, we obtain:
xm(cx'' + (m+1)cx') - 6mx(cx' / x) = 0
Simplifying and factoring out c[tex]x^m[/tex], we get:
c[tex]x^m[/tex][(m+1)(m) - 6m] = 0
Since x > 0, we can divide both sides by c[tex]x^m[/tex] to get:
[tex]m^2[/tex] - 5m = 0
Solving for m, we get:
m = 0 or m = 5
Therefore, the general solution of the homogeneous equation is:
yh' = c₁ + c₂[tex]x^5[/tex]
Next, we need to find a particular solution of the non-homogeneous equation. We assume that the solution has the form:
y'p = u₁(x)y₁(x) + u₂(x)y₂(x)
where y₁ and y₂ are linearly independent solutions of the homogeneous equation (in this case, y₁ = 1 and y₂ = [tex]x^5[/tex]), and u₁ and u₂ are functions to be determined.
We differentiate y'p to obtain:
y'p' = u₁'y₁ + u₂'y₂ + u₁y₁' + u₂y₂'
and
y'p'' = u₁''y₁ + u₂''y₂ + 2u₁'y₁' + 2u₂'y₂' + u₁y₁'' + u₂y₂''
Substituting these into the non-homogeneous equation and simplifying, we get:
u₂''(x)[tex]x^5[/tex] - 6u₂'(x)[tex]x^4[/tex] = [tex]x^6[/tex]
We can solve this differential equation for u₂(x) using the method of undetermined coefficients, by assuming that u₂(x) has the form:
u₂(x) = A[tex]x^2[/tex] + Bx + C
where A, B, and C are constants to be determined. Substituting this into the above equation and equating coefficients of like terms, we get:
A = 0, B = 0, and C = -1/18
Therefore, the particular solution is:
yp' = -(1/18)[tex]x^2[/tex]
The general solution of the non-homogeneous equation is then:
y = yh' + yp' = c₁ + c₂[tex]x^5[/tex]- (1/18)[tex]x^2[/tex]
where c₁ and c₂ are constants determined by the initial or boundary conditions.
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a queuing system has three servers with expected service times of 5 minutes, 30 minutes, and 10 minutes. the service times are exponentially distributed. each server has been busy with a current customer for 5 minutes. determine the expected remaining time until the next service completion. that is, what is the expected waiting time?
The answer is that the expected waiting time until the next service completion is 7.44 minutes.
To calculate the expected waiting time, we need to first find the expected remaining service time for each server. Since the service times are exponentially distributed, the expected remaining service time for each server is equal to the reciprocal of its service rate. The service rate is the inverse of the expected service time.
Thus, the expected remaining service time for the first server is 1/λ1 = 1/0.2 = 5 minutes, where λ1 is the arrival rate for the first server. Similarly, the expected remaining service time for the second and third servers are 1/λ2 = 1/0.0333 = 30 minutes and 1/λ3 = 1/0.1 = 10 minutes, respectively.
Since each server has been busy for 5 minutes with a current customer, the expected remaining service time for each server is reduced by 5 minutes. Thus, the expected remaining service times are 0 minutes, 25 minutes, and 5 minutes, respectively.
The expected waiting time until the next service completion is equal to the sum of the expected remaining service times weighted by the probability that a customer arrives at each server while it is busy. The probability of a customer arriving at each server while it is busy can be calculated using the Erlang C formula.
Using the Erlang C formula, we can calculate that the probability of a customer arriving at the first server while it is busy is 0.016, the probability of a customer arriving at the second server while it is busy is 0.524, and the probability of a customer arriving at the third server while it is busy is 0.091.
Thus, the expected waiting time until the next service completion is (0 minutes)*(0.016) + (25 minutes)*(0.524) + (5 minutes)*(0.091) = 7.44 minutes.
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The average weight of a high school freshman is 142 pounds. If a sample of twenty
freshmen is selected, find the probability that the mean of the sample will be greater than
145 pounds. Assume the variable is normally distributed with a standard deviation of 12.3
pounds.
The probability that the mean weight of a sample of twenty freshmen will be greater than 145 pounds is 0.138.
What is the probability?The probability is determined using the central limit theorem and the formula for the standard error of the mean:
SE = σ/√nwhere;
SE is the standard error of the mean,σ is the population standard deviation, andn is the sample size.Data given;
σ = 12.3 pounds; n = 20
SE = 12.3/√20
SE = 2.75 pounds.
The sample mean is then standardized using the z-score formula:
z = (x - μ) / SE
z = (145 - 142) / 2.75
z = 1.09
Using a calculator, the probability of a z-score greater than 1.09 is 0.138.
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contaminated water: the concentration of benzene was measured in units of milligrams per liter for a simple random sample of five specimens of untreated wastewater produced at a gas field. the sample mean was with a sample standard deviation of . seven specimens of treated wastewater had an average benzene concentration of with a standard deviation of . it is reasonable to assume that both samples come from populations that are approximately normal. can you conclude that the mean benzene concentration is less in treated water than in untreated water? let denote the mean benzene concentration for untreated water and denote the mean benzene concentration for treated water. use the level the -value method with the ti-84 plus calculator.
The answer is that we can use a hypothesis test to determine if the mean benzene concentration is less in treated water than in untreated water.
: We will use the following hypotheses:
H0: μtreated ≥ μuntreated
Ha: μtreated < μuntreated
where μtreated is the true mean benzene concentration for treated water, and μuntreated is the true mean benzene concentration for untreated water.
We will use a one-tailed t-test with a level of significance of α = 0.05.
Using the Ti-84 Plus calculator, we can find the t-test statistic and p-value. First, we calculate the pooled standard deviation:
Sp =√((n1-1)*s1² + (n2-1)*s2²)/(n1+n2-2))
Sp = √(((5-1)*0.7² + (7-1)*0.6²)/(5+7-2))
Sp = 0.642
Next, we calculate the t-test statistic:
t = (x1 - x2) / (Sp * √(1/n1 + 1/n2))
t = (0.5 - 0.8) / (0.642 *√(1/5 + 1/7))
t = -2.13
Finally, we calculate the p-value:
p-value = tcdf(-100, t, df)
p-value = tcdf(-100, -2.13, 10)
p-value = 0.026
Since the p-value is less than the level of significance, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the mean benzene concentration is less in treated water than in untreated water.
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find the p-value based on a standard normal distribution for the standardized test statistic and provided alternative hypothesis.
z= -1.86 for Ha: p <0.5
The p-value for a standardized test statistic of z = -1.86 with the alternative hypothesis Ha: p < 0.5 is 0.0322.
To find the p-value for the standardized test statistic of z = -1.86 with the alternative hypothesis Ha: p < 0.5, we need to find the area under the standard normal distribution curve to the left of z = -1.86.
We can use a standard normal distribution table or a calculator to find this area. Using a calculator, we can use the following steps:
1) Calculate the cumulative distribution function (CDF) of the standard normal distribution at z = -1.86. This gives us the area under the curve to the left of z = -1.86.
CDF(-1.86) = 0.0322
2) Since the alternative hypothesis is one-tailed (p < 0.5), we need to find the area in the left tail of the standard normal distribution. Therefore, the p-value is the same as the area under the curve to the left of z = -1.86.
p-value = 0.0322
So the p-value for a standardized test statistic of z = -1.86 with the alternative hypothesis Ha: p < 0.5 is 0.0322.
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Think about if you solve an equation and at the end you are left with the equaton 9=15 What does this mean?
When you are left with the equation "9=15" after solving an equation, it means that there is no solution that satisfies the equation.
Now, if you have solved an equation and ended up with the statement "9=15," this means that the equation is not true. In mathematical terms, the equation "9=15" is a contradiction or an inconsistent equation. This is because it is impossible for the value of 9 to be equal to the value of 15.
When solving an equation, it is important to check the solution by substituting the value back into the original equation to ensure that it satisfies the equation. If the solution leads to a contradiction like "9=15," then the solution is invalid and does not satisfy the equation.
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at travis' birthday party, `\frac{3}{4}` of his birthday cake was eaten. the next day, travis ate `\frac{1}{3}` of the remaining cake. what fraction of the whole cake did travis eat the next day
Travis ate 1/12 of the whole cake the next day
A fraction represents a part of a whole. In this case, the whole cake represents the whole, and the part that was eaten represents the fraction. When we say that 3/4 of the cake was eaten, it means that out of the whole cake, 3/4 or three-fourths of the cake was consumed.
Travis ate 1/3 of the remaining cake the next day.
This means that after 3/4 of the cake was eaten, there was 1/4 of the cake remaining. Travis ate 1/3 of that remaining 1/4 of the cake, which can be written as
=> 1/3 x 1/4.
To simplify this fraction, we multiply the numerators (1 x 1) and the denominators (3 x 4), giving us 1/12.
We can write this fraction as a percentage, which is 8.33%. To summarize, fractions are used to represent parts of a whole, and in this case, Travis ate 1/12 or 8.33% of the whole cake the next day.
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The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives 16 1616 years; the standard deviation is 1. 7 1. 71, point, 7 years. Use the empirical rule ( 68 − 95 − 99. 7 % ) (68−95−99. 7%)left parenthesis, 68, minus, 95, minus, 99, point, 7, percent, right parenthesis to estimate the probability of a gorilla living between 14. 3 14. 314, point, 3 and 19. 4 19. 419, point, 4 years
Using the empirical rule, the estimated probability of a gorilla living between 14.3 and 19.4 years is 95
We have,
Find the z-scores corresponding to the values and then use the empirical rule percentages.
The formula for the z-score is:
z = (X - μ) / σ
where X is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation.
Find the z-score for X = 14.3 years.
= (14.3 - 16) / 1.71 ≈ -1.05
Find the z-score for X = 19.4 years.
= (19.4 - 16) / 1.71 ≈ 1.76
Use the empirical rule percentages to estimate the probability of a gorilla living between 14.3 and 19.4 years.
For the interval between 14.3 and 19.4 years, we are interested in the area between -1.05 and 1.76 on the normal distribution curve.
The empirical rule percentages are:
68% of the data falls within 1 standard deviation from the mean.
95% of the data falls within 2 standard deviations from the mean.
99.7% of the data falls within 3 standard deviations from the mean.
Since the z-scores for 14.3 and 19.4 are within 2 standard deviations from the mean (-1.05 and 1.76), we can estimate that approximately 95% of the gorillas' lifespans in the zoo fall between 14.3 and 19.4 years.
Thus,
Using the empirical rule, the estimated probability of a gorilla living between 14.3 and 19.4 years is 95
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The complete question:
"The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives 16.16 years, and the standard deviation is 1.71 years. Use the empirical rule (68% - 95% - 99.7%) to estimate the probability of a gorilla living between 14.3 and 19.4 years."
James has a triangle with a perimeter of 12. The triangle is dilated with a scale factor of 3. What is the new perimeter?
The new perimeter of the triangle after it is dilated with a scale factor of 3 is 36 units.
If James has a triangle with a perimeter of 12, it means that the sum of the lengths of all three sides of the triangle is 12. Let's call the lengths of the three sides a, b, and c, where a + b + c = 12.
When the triangle is dilated with a scale factor of 3, it means that all the sides of the triangle are multiplied by 3. Let's call the new lengths of the sides A, B, and C, where A = 3a, B = 3b, and C = 3c.
The new perimeter of the triangle is the sum of the lengths of the new sides, which is:
A + B + C = 3a + 3b + 3c
We know that a + b + c = 12, so we can substitute this into the above equation:
A + B + C = 3(12)
A + B + C = 36
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Some members of a community garden in California want to plant an orchard to earn some extra income. After researching, they decided to plant avocado trees. Avocado saplings (baby trees) cost $20 each. It takes 3 years for avocado trees to reach maturity and bear fruit, but after they do, each tree will produce $125 worth of fruit. The community garden is made of 50 members and their goal is to sell $250 per capita each year.
Calculate the total number of trees that all the garden members will need in total in their orchard to meet the goal.
The total number of trees that all the garden members will need in total in their orchard to meet the goal is 40 trees.
To meet the goal of selling $250 per capita each year, the community garden will need to generate a total of:
[tex]$250 * 50 members[/tex] = [tex]$12,500 per year[/tex]
Each avocado tree costs $20 and produces $125 worth of fruit per year after maturity. Therefore, the net revenue per tree per year is:
$[tex]125[/tex]- $[tex]20[/tex] = $[tex]105[/tex]
Since it takes 3 years for a tree to mature, we can calculate the net revenue per tree over 3 years as:
$[tex]105[/tex] x [tex]3[/tex] = $[tex]315[/tex]
To meet the annual revenue goal of $12,500, the community garden will need to plant:
$[tex]12,500[/tex] / $315 per 3-year period = [tex]39.68[/tex] trees
Since we can't plant fractional trees, we need to round up to the nearest whole number. Therefore, the total number of trees that all the garden members will need in total in their orchard to meet the goal is: 40 trees
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for a bill totalling $5.65, the cashier received 25 coins consisting of nickels and quarters. how many nickels did the cashier receive?
Answer: 3 Nickles.
Step-by-step explanation:
22 quarters adds up to $5.50
The remaining 15c is accounted by the last 3 coins, which are nickles.
Erin and Shelby have 8 children who never finish their dinner. Tonight they are having soup. Calculate how much soup is left over after everyone has finished eating. Erin and Shelby ate all their soup. Three of their kids left 3/4 cup of soup in their bowl. Two of their kids left 1/4 cup of soup in their bowl and three of their kids left 1/2 cup of soup in their bowl. How much soup is left over?
Answer:
Step-by-step explanation:
Let's start by finding out how much soup was initially in the pot. We know that Erin, Shelby, and all 8 of their children ate some soup, but we don't know how much.
If we add up the amounts left in the bowls, we can find out how much soup they didn't eat:
3 kids left 3/4 cup each = 3 * 3/4 = 9/4 cups
2 kids left 1/4 cup each = 2 * 1/4 = 1/2 cup
3 kids left 1/2 cup each = 3 * 1/2 = 3/2 cups
Adding these amounts together:
9/4 + 1/2 + 3/2 = 5 cups
So they left 5 cups of soup in their bowls.
If we assume that each person had one serving of soup (even though some left some in their bowls), and that each serving was the same size, then the amount of soup they didn't eat is equal to the amount of soup that was left in the pot.
So, the amount of soup left over is 5 cups.
A construction company borrowed$75,000 for 4 months at an annual interest rate 8%. Find the simple interest due on the loan
Answer:
The answer is SI of $1980
