if the sales ax rate is 7.6%, how much sales tax will you pay on a $330 item?

Answer:208.75

Step-by-step explanation:

The length of the diagonal SU will be 46 units.

A parallelogram is a quadrilateral with two pairs of equal sides.

Given is a parallelogram RSTU.

The diagonals of a parallelogram bisect each other. This means that -

SV = VU

2x + 3 = 4x - 17

17 + 3 = 4x - 2x

20 = 2x

x = 10

The length SU will be -

SU = SV + VU

SU = 2x + 3 + 4x - 17

SU = 6x - 14

SU = 60 - 14

SU = 46 units

Therefore, the length of the diagonal SU will be 46 units.

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C)The statement is false. The definition of AB states that each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.(C)

To see why this is the case, let A be an m x n matrix and let B be an n x p matrix. Then the product AB is an m x p matrix whose (i,j)-th entry is given by the dot product of the i-th row of A with the j-th column of B:

(AB){i,j} = \sum{k=1}^n a_{i,k} b_{k,j}

This means that the j-th column of AB is obtained by taking a linear combination of the columns of B using the weights from the j-th column of A.

In other words, the statement "each column of AB is a linear combination of the columns of B using weights from the corresponding column of A" is true, but the statement in answer choice C is false.

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The expression of a function is
(a) {"++", "+-", "-+", "--"}

(b) {"000", "001", "010", "011", "100", "101", "110", "111"}

(a) Roster notation is used to express a set of elements in a compact form. In this case, A is the set {+, -}, and A^2 is the set of all possible ordered pairs of elements from A. This set can be expressed using roster notation as {"++", "+-", "-+", "--"}. The double symbols indicate the two elements of each ordered pair.

(b) Here, A is the set {0, 1}, and A^3 is the set of all possible ordered triplets of elements from A. This set can be expressed using roster notation as {"000", "001", "010", "011", "100", "101", "110", "111"}. The triple symbols indicate the three elements of each ordered triplet.

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The number of students who went to Camp Redwood was 23% more than the number of students who went to Camp Arrowhead.

What is percentage ?

A percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is "%". For example, 50% means 50 out of 100 or 0.5 as a decimal.

Percentages are commonly used to represent proportions, rates, and changes in quantities. They can be used to compare quantities of different sizes, or to express how much one quantity has changed relative to another quantity.

Given by the question:
Let's assume that there are a total of 100 students who were surveyed.

If 15% of the students who planned to go to Camp Redwood did not go, then 85% of them did go:

0.85 x (number of students who planned to go to Camp Redwood) = number of students who actually went to Camp Redwood

Similarly, if 20% of the students who planned to go to Camp Arrowhead did not go, then 80% of them did go:

0.80 x (number of students who planned to go to Camp Arrowhead) = number of students who actually went to Camp Arrowhead

Let's say that x students planned to go to Camp Redwood and y students planned to go to Camp Arrowhead. Then we can write two equations based on the above percentages:

0.85x = number of students who actually went to Camp Redwood

0.80y = number of students who actually went to Camp Arrowhead

We want to find the difference between the number of students who went to Camp Redwood and the number of students who went to Camp Arrowhead, so we can subtract the second equation from the first:

0.85x - 0.80y = (number of students who actually went to Camp Redwood) - (number of students who actually went to Camp Arrowhead)

We can substitute the values from the percentages into the equation:

0.85x - 0.80y = 0.85(0.85x) - 0.80(0.80y)

0.85x - 0.80y = 0.7225x - 0.64y

0.13x = 0.16y

We can solve for one of the variables in terms of the other:

x = (0.16/0.13)y

x = 1.23y

So if y students planned to go to Camp Arrowhead, then 1.23y students planned to go to Camp Redwood. The difference between the number of students who went to Camp Redwood and the number of students who went to Camp Arrowhead is:

1.23y - y = 0.23y

So the number of students who went to Camp Redwood was 23% more than the number of students who went to Camp Arrowhead.

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The sample mean of the given frequency distribution of heart rates is 77.59 and the sample variance is 27.543.

To calculate the sample mean and sample variance of the given frequency distribution of heart rates, we must first calculate the mid-points of each class (i.e. 63.5, 69.5, 76.5, 82.5, 87.5). Then, we must calculate the mid-point multiplied by the frequency of each class [tex](4x63.5, 8x69.5, 5x76.5, 7x82.5, 13x87.5)[/tex]. Finally, we must add up all of these values together and divide by the total frequency. This gives us the sample mean.

To calculate the sample variance, we must first calculate the squared differences of each midpoint multiplied by the frequency from the sample mean [tex](4x63.5, 8x69.5, 5x76.5, 7x82.5, 13x87.5)[/tex]. Then, we must add up all of these values together and divide by the total frequency minus one. This gives us the sample variance.

Therefore, the sample mean of the given frequency distribution of heart rates is 77.59 and the sample variance is 27.543.

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If the student is selected randomly, the probabilities for the following is given as:

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution. Knowing the total number of outcomes is necessary before we can calculate the likelihood that a specific event will occur.

P (A1∩B1) = 5/43 = 0.1163

P(A1∪B1) = 14/43 + 22/43 - 5/43 = 31/43 = 0.7209

P(A1|B1) = 5/22 = 0.2273

P(B2'∪A1') = (1-21/43) + (1-14/43) - 26/43

= 25/43 = 0.2814

P(A2∩B1) = 17/43 = 0.3953

P(A2∩B2)= 12/43 = 0.2791

P(A2∩B1) is more i.e. student with left thumb on top is more .Hence, student selects left thumb on top.

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Complete question:

Let A1 and A2 be the events that a person is left-eye dominant or right-eye dominant, respectively. When a person folds his/her hands, let B, and B2 be the vents that the left thumb and right thumb, respectively, are on top. A survey in one statistics class yielded the table below: B B2 Totals A 5 9 14 17 12 29 Totals 21 If a student is selected randomly, find the following probabilities: a) P(A, NB) b) P(AUB) c) P(A1B1) d) P(B' UA) e) If the students had their hands folded and you hoped to select a right-eye-dominant student, would you select a "right thumb on top” or a “left thumb on top” student? Why?

The final measures of forecast accuracy are: CSE = -46, MAD = 18.4, MSE = 462.8, MAPE = 12.44

The forecast errors can be calculated as follows:

Period 1: 751 - 751 = 0

Period 2: 741 - 751 = -10

Period 3: 728 - 749 = -21

Period 4: 773 - 745 = 28

Period 5: 718 - 751 = -33

The cumulative sum of the forecast errors (CSE) is: 0 - 10 - 21 + 28 - 33 = -46

The mean absolute deviation (MAD) can be calculated as follows:

MAD = (abs(0) + abs(-10) + abs(-21) + abs(28) + abs(-33)) / 5

MAD = (0 + 10 + 21 + 28 + 33) / 5

MAD = 92 / 5 = 18.4

The mean squared error (MSE) can be calculated as follows:

MSE = (0^2 + (-10)^2 + ( -21)^2 + 28^2 + (-33)^2) / 5

MSE= ( 0+ 100+ 441 + 784 + 1089)/5

MSE = 2314/ 5 = 462.8

The mean absolute percent error (MAPE) can be calculated as:

MAPE = (abs(0/751) + abs(-10/741) + abs(-21/728) + abs(28/773) + abs(-33/718)) / 5 * 100

MAPE = (0 + 1.34 + 2.89 + 3.63 + 4.58) / 5 * 100

MAPE = 12.44

The final measures of forecast accuracy are:

CSE = -46

MAD = 18.4

MSE = 462.8

MAPE = 12.44

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_______The given question is incorrect, the correct question is given below:

The realized demand values A= 751, 741, 728, 773, 718

and forecast values F = 751,  751, 749, 745, 751

for periods = 1,2,3,4,5

are provided in the table below. Round only the final measure of forecast accuracy to two decimals. The cumulative sum of the forecast errors (CSE) is (Round to two decimals) The mean absolute deviation value (MAD) is (Round to two decimals) The mean squared error value (MSE) is (Round to two decimals) The mean absolute percent error value (MAPE) is (Round to two decimals.)

Answer:

The inequality to set up is [tex]t+16 \ge 31[/tex]

It solves to [tex]t \ge 15[/tex]

Bill needs to run at least 15 more miles.

================================================

Explanation:

He has run 16 miles so far. Add on another t miles to get 16+t or t+16 to represent the total amount he runs. This expression must be 31 or larger

Either t+16 > 31 or t+16 = 31

Those two items condense to [tex]t+16 \ge 31[/tex]

To solve for t, we subtract 16 from both sides to undo the +16.

[tex]t+16 \ge 31\\\\t+16-16 \ge 31-16\\\\t \ge 15\\\\[/tex]

Bill needs to run at least 15 more miles.

Meaning that t = 15, t = 16, t = 17, etc.

The sample variance and standard deviation are given as follows:

To obtain the variance and the standard deviation, first we must obtain the mean, which is given by the sum of all observations divided by the number of observations, hence:

Mean = (77.1 + 82.9 + 90 + 91.3 + 81.9 + 83.3 + 84.9 + 82.5 + 84.6)/10

Mean = 75.85.

Then we must obtain the sum of the differences squared between each observation and the mean, as follows:

(77.1 - 75.85)² + (82.9 - 75.85)² + (90 - 75.85)² + (91.3 - 75.85)² + (81.9 - 75.85)² + (83.3 - 75.85)² + (84.9 - 75.85)² + (82.5 - 75.85)² + (84.6 - 75.85)² = 784.9825

The sample variance is given by the above sum divided by one less than the sample size, hence:

Variance = 784.9825/9

Variance = 87.22 %².

The standard deviation is the square root of the variance, hence:

Standard deviation = sqrt(87.22)

Standard deviation = 9.34%.

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The graph of the functions f(x) and g(x) on the same coordinates plane and the solution is (0 , 0.818).

What is a function?

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

Two functions:

f(x) = -|x - 3| + 2

g(x) = 12x - 3

Now,

The graph is given below.

The intersection of the graph of f(x) and g(x) is the solution to f(x) = g(x).

So,

for x> 3 , we can say that

-|x - 3| + 3 = 12x -3

-x + 3 +3 = 12x -3

Therefore , x = 9/11 = 0.818

for x< 3 , we can say that

-|x - 3| + 3 = 12x -3

x - 3 = 12x - 3

x = 0

The solution is (0 , 0.818).

Thus, the solution is (0 , 0.818).

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Does the data give evidence of an association between scoring at least 21 points and the football team winning the game?

Ans. There is a weak, negative association.

A weak, negative association refers to a relationship between two variables where an increase in one variable is associated with a decrease in the other variable, but the association is not strong.

In statistics, the strength of the association between two variables is measured by a correlation coefficient.

A correlation coefficient ranges from -1 to 1, where -1 represents a perfect negative correlation, 0 represents no correlation, and 1 represents a perfect positive correlation.

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The sign that makes the statement true is 792 3/20 ≠ 792 1/2.

A fraction is written in the form of p/q, where q ≠ 0.

There are two types of fractions: proper fractions in which the numerator is less than the denominator and,

Improper fractions in which the numerator is larger than the denominator.

Given, Are two fractions 792 3/20 and 792 1/2.

The sign that makes the statement true is 792 3/20 ≠ 792 1/2.

Some other signs, for example, Inequalities can also make the statement true.

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The probability of waiting less than 12 minutes between successive speeders is 0.02.

The time elapsed between successive speeders detected by a radar unit is a  random variable with a cumulative distribution function:

F(x) = 0 when x = 0.

x/10, for 0 x ten 1, for x ten

a) Using the cumulative distribution function, to calculate the probability of waiting less than 12 minutes (0.2 hours) between successive speeders: F(0.2):

F(0.2) = 0.2/10 = 0.02

So, the probability of waiting less than 12 minutes between speeders is 0.02.

b) Using the probability density function, we can differentiate the cumulative distribution function with respect to x to find the probability:

For x = 0, f(x) = dF(x)/dx = 0.

1/10 for 0 x 10 0, 0 for x 10

The area under the probability density function up to x=0.2 represents the probability of waiting less than 12 minutes:

P(0 < w < 0.2) = ∫0.2 0 f(x) (x) dx = ∫0.2 0 (1/10) dx = (1/10) * [x] 0.2 = 0.02

Using either the cumulative distribution function or the probability density function, the probability of waiting less than 12 minutes between successive speeders is 0.02.

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A mathematical statement known as an equation is created by joining two expressions with an equal sign. For instance, 3x - 5 = 16 is an equation. This equation may be solved, and the result shows that the value of the variable x is 7.

Given,

(6.82x - 2.53y - 5.72) - (-5.72 + 9.45y - 8.11 x)

Simplifying the above equation,

= 6.82x - 2.53y - 5.72 - (-5.72 + 9.45y - 8.11 x)

= 6.82x - 2.53y - 5.72 + 5.72 - 9.45y + 8.11 x

Simplify

6.82x - 2.53y - 5.72 + 5.72 - 9.45y + 8.11 x : 14.93x - 11.98y

Then we get,

= 14.93x - 11.98y

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The limit of 10 pounds per shipment of medium-sized box and the 1 pound weight per jeans and 2 pound weight per pair of shoes, indicates that the graph and ordered pair of inequality are as follows;

13. Please find attached the required graph of the inequality, created with MS Excel

14. 5 pairs of jeans and 4 pairs of shoes weigh more than 10 pounds and can not be packaged in the same medium box.

An inequality is relationship that compares expressions with different values.

13. The inequality in the question, x + 2·y ≤ 10, indicates that we get;

x + 2·y ≤ 10

2·y ≤ 10 - x

y ≤ (10 - x)/2 = 5 - x/2

y ≤ 5 - x/2

The graph of the inequality, x + 2·y ≤ 10 can be created using the above inequality. Please find attached the required graph created using with the inequality expressed as an equation, followed by shading the feasible region, using MS Excel.

14. The point on the graph that corresponds to 5 pairs of jeans and 4 pairs of shoes, (5, 4), is not in the feasible region. Based on the point (5, 2.5) on the graph, when 5 pairs of jeans are packed in the box, only 2 pairs of shoes can be included. Therefore;

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The value of g(x) = [x] At, 2 ≤ x < 3, g(x) = 2, It is also known as the greatest integer function.

A function on real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals.

Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

A step function f(x) = [x] is called a step function where the output is the greatest integer less than or equal to the input.

The function is g(x) = [x].

At, - 1 ≤ x < 0, g(x) = 0.

At, 1 ≤ x < 2, g(x) = 1.

At, 2 ≤ x < 3, g(x) = 2.

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The value of the d(g o f)/dt at t = 2 is 14.

We can start by computing the composition (g o f)(t) and then finding its derivative with respect to t using the chain rule.

(g o f)(t) = g(f(t)) = g(t, t^2 - 4e^(t-2))

At t = 2, we have f(2) = 2 and f'(2) = 2t - 4e^(t-2) evaluated at t = 2 gives f'(2) = 0.

To find d(g o f)/dt at t = 2, we first need to evaluate (g o f)(2):

(g o f)(2) = g(f(2)) = g(2, 0) = g(a)

Next, we use the chain rule to find the derivative of (g o f) with respect to t:

[tex](d/dt) (g o f)(t) = (∂g/∂x)(a) (df/dt) + (∂g/∂y)(a) (d/dt) (t^2 - 4e^{(t-2)}) + (∂g/∂z)(a) (dg/dz)(a)[/tex]

Since f(t) = t, df/dt = 1. Also, since g(x,y,z) is differentiable, we can write dg/dz(a) as ∂g/∂z(a) = 2.

Substituting the values we have and evaluating at t = 2, we get:

[tex](d/dt) (g o f)(2) = (4)(1) + (2)(2t - 4e^{(t-2)})(t=2) + (2)(2)[/tex]

= 4 + 2(4-4e^0) + 4

= 14

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Work Shown:

ST + TU = SU .... segment addition postulate

(4x-4) + (4x-10) = 5x+1

8x-14 = 5x+1

8x-5x = 1+14

3x = 15

x = 15/3

x = 5

Then use this x value to determine the segment lengths.

As a check: ST+TU = 16+10 = 26 which matches with SU = 26. This confirms ST+TU = SU is the case and it confirms the final answer.

The option (b) set y equal to a constant other than zero to get the curve of points where the slope is that constant.

A slope field is a visual representation of the slopes of a function's derivative at different points on the xy-plane. The slope at a particular point in the xy-plane is given by the derivative of the function at that point.

To match the given equations with their corresponding slope fields, you need to analyze the behavior of the slopes and isoclines. An isocline is a curve in the plane along which the slope of a function is constant. Isoclines are useful in constructing slope fields by hand, as they help in determining the slope at different points.

To start with, set y equal to zero and observe how the derivative behaves along the x-axis. Similarly, set x equal to zero and observe the behavior of the derivative along the y-axis. These observations will give you an idea of the direction and magnitude of the slope at different points in the xy-plane.

Next, consider the curve in the plane defined by setting y' = 0. This curve represents the points in the picture where the slope is zero. These points are known as critical points or equilibrium points. They are essential in analyzing the stability and behavior of a system.

These curves are called isoclines and can be used to construct the slope field picture by hand.

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Complete Question:

Match the following equations with their slope field. Clicking on each picture will give you an enlarged view. While you can probably sove this problem by guessing, it is useful to try to predict characteristics of the slope field and then match them to the picture Here are some handy characteristics to start with- you will develop more as you practice. Set y equal to zero and look at how the derivatiwe behaves along the x axis. Do the same for the y axis by setting x equal to 0. Consider the curve in the plane defined by setting y' = 0-this should correspond to the points in the picture where the slope is zero. Setting y equal to a constant other than zero gives the curve of points where the slepe is that constant. These are called isoclines, and can be used to construct the slope field picture by hand.

For example, 0.123 would be a legitimate entry. Lower limit (first blank) = 0.285, Upper limit (second blank) = 0.562.

The confidence interval for proportions is

p±z *√p(1-p)/n.

In this case, p = 36/85 = .4235, and z* = 2.575 (the value of the standard normal distribution corresponding to 99%).

so,

0.4235 ± 2.576 * √0.4235*0.5765/85

0.4235 ± 0.13804

= (0.285, 0.562)

Therefore, Lower limit is 0.285 and Upper limit 0562.

A confidence interval is the mean of your estimate plus and minus the variation in that estimate. This is the range of values you anticipate your estimate to fall between if you redo your test, within a certain position of confidence.

Confidence, in statistics, is another way to describe probability. For illustration, if you construct a confidence interval with a 95 confidence position, you're confident that 95 out of 100 times the estimate will fall between the upper and lower values specified by the confidence interval.

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The probability that exactly 90 defective washers are found is (400 choose 90) * (1100 choose 110) / (1500 choose 200). and the probability that at least 2 defective items are found is (400 choose 1) * (1100 choose 199) / (1500 choose 200).

Let X be the number of defective washers in a sample of 200 washers.

We can model X as a hypergeometric distribution with parameters N = 1500 (total number of washers), K = 400 (total number of defective washers), and n = 200 (sample size).

a) The probability of finding exactly 90 defective washers is:

P(X = 90) = (400 choose 90) * (1100 choose 110) / (1500 choose 200)

This is because we need to choose 90 defective washers from the 400 defective washers, and 110 non-defective washers from the 1100 non-defective washers, out of the total of 200 washers chosen.

b) The probability of finding at least 2 defective items can be calculated as the complement of the probability of finding 0 or 1 defective item in the sample:

P(X >= 2) = 1 - P(X = 0) - P(X = 1)

To compute P(X = 0), we need to choose 0 defective washers from the 400 defective washers, and 200 non-defective washers from the 1100 non-defective washers, out of the total of 1500 washers:

P(X = 0) = (400 choose 0) * (1100 choose 200) / (1500 choose 200)

To compute P(X = 1), we need to choose 1 defective washer from the 400 defective washers, and 199 non-defective washers from the 1100 non-defective washers, out of the total of 1500 washers:

P(X = 1) = (400 choose 1) * (1100 choose 199) / (1500 choose 200)

Once we have computed P(X = 0) and P(X = 1), we can substitute these values into the expression for P(X >= 2) to obtain the desired probability.

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The polynomial degree of 12x^4 - 8x + 4x^2 - 3 is option C. 4 .

The highest or greatest power of a variable in a polynomial equation is referred to as the degree of the polynomial. The degree denotes the polynomial's highest exponential power (ignoring the coefficients).

The highest degree of a polynomial's monomials with non-zero coefficients is referred to as the polynomial's degree in mathematics. A term's degree is a non-negative integer and is calculated by adding the exponents of all the variables that occur in it.

To know the degree we will arrange as

12x^4 - 8x + 4x^2 - 3

12x^4+ 4x^2 - 8x  - 3

Then we can see that the highest degree is 4.

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Answer:

the solution to the differential equation is:

y + (1/3)y^3 = e^x + 1/3.

Step-by-step explanation:

We can use the equation e^x=(1/(1+y^2))dy/dx to solve this differential equation using separation of variables.

First, we can rewrite the equation as:

(1+y^2)dy = e^x dx

Next, we can separate the variables:

(1+y^2)dy = e^x dx

∫ (1+y^2)dy = ∫ e^x dx

y + (1/3)y^3 = e^x + C

where C is the constant of integration.

Now we can use the initial condition to solve for C. Let's say the initial condition is y(0) = 1, then we have:

1 + (1/3)(1)^3 = e^0 + C

4/3 = 1 + C

C = 1/3

Therefore, the solution to the differential equation is:

y + (1/3)y^3 = e^x + 1/3.

The z-scores for the given data values  520, 640, 500, 470, and 320 are:

0.2, 1.4. 0, -0.3, -1.8

To calculate the z-scores for each data value, we can use the formula:

z = (x - μ) / σ

where:

x is the data value

μ is the population mean (in this case, 500)

σ is the population standard deviation (in this case, 100)

Let's calculate the z-scores for each data value:

For x = 520:

z = (520 - 500) / 100 = 0.2

For x = 640:

z = (640 - 500) / 100 = 1.4

For x = 500:

z = (500 - 500) / 100 = 0

For x = 470:

z = (470 - 500) / 100 = -0.3

For x = 320:

z = (320 - 500) / 100 = -1.8

Therefore, the z-scores for the given data values are:

z-score for 520: 0.2

z-score for 640: 1.4

z-score for 500: 0

z-score for 470: -0.3

z-score for 320: -1.8

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The value of c is given as follows:

c = 23.

The problem states that the total of 13 cherries, 8 more cherries, and 2 more cherries is c, hence the value of c is obtained with the addition of these amounts, as follows:

c = 13 + 8 + 2.

Adding these three amounts, the numeric value of c is given as follows:

c = 23.

The problem asks for the numeric value of c.

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The correct option is (d): △ABC is congruent to △A′B′C′ because you can map △ABC to △A′B′C′ using a translation 6 units to the right, which is a rigid motion.

In option (a), a reflection across the y-axis is not a rigid motion, as it changes the orientation of the triangles, and therefore cannot be used to show congruence.

In option (b), if there is no sequence of rigid motions that maps △ABC to △A′B′C′, then the triangles are not congruent.

In option (c), a translation 6 units to the left does not map △ABC to △A′B′C′. However, a translation 6 units to the right would map △ABC to △A′B′C′, because translations are rigid motions that preserve distance and angle measures.

Therefore, option (d) is the correct statement that describes the relationship between the two triangles.

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The statement "parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals." is proved.

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are congruent, meaning they have the same length. Similarly, the opposite angles of a parallelogram are congruent, meaning they have the same measure.

Now, let's consider a parallelogram ABCD with sides AB, BC, CD, and DA. We want to show that:

AB² + BC² + CD² + DA² = AC² + BD²

where AC and BD are the diagonals of the parallelogram.

First, let's draw the diagonals AC and BD. This divides the parallelogram into four triangles: triangle ABC, triangle BCD, triangle CDA, and triangle DAB.

Next, let's use the Pythagorean theorem in each of these triangles. Recall that the Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

In triangle ABC, we have:

AC² = AB² + BC²

Similarly, in triangle CDA, we have:

AC² = CD² + DA²

Adding these two equations together, we get:

2AC² = AB² + 2BC² + CD² + 2DA²

Now, let's look at triangle ABD. We have:

BD² = AB² + DA²

Similarly, in triangle BCD, we have:

BD² = BC² + CD²

Adding these two equations together, we get:

2BD² = AB² + 2BC² + CD² + 2DA²

Finally, adding the two equations we obtained for AC² and BD², we get:

2AC² + 2BD² = 2AB² + 4BC² + 2CD² + 2DA²

Simplifying this expression, we get:

AC² + BD² = AB² + BC² + CD² + DA²

Therefore, we have proven that in any parallelogram, the sum of the squares of the lengths of the four sides is equal to the sum of the squares of the lengths of the two diagonals.

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a) The joint probability density function is [tex]f(y_1,y_2) = f(y_1) * f(y_2) = (1/3e^{(-y1/3)})(1/3e^{-y2/3})[/tex]

b) The total effective length of life of the two fuses is less than or equal to one.

The exponential distribution is a probability distribution that models the length of life of fuses, and its probability density function can be used to find the joint probability density function for the lengths of two independent fuses.

a) To find the joint probability density function for the lengths of two independent fuses, we simply multiply the probability density functions of each individual fuse. In this case, we have

[tex]= > f(y_1,y_2) = f(y_1) * f(y_2) = (1/3e^{(-y1/3)})(1/3e^{-y2/3})[/tex]

for y₁, y₂ > 0. This is a function that gives the probability of a given pair of lengths (y₁,y₂) occurring.

b) To find P(Y₁ + Y₂ ≤ 1), we must first determine the cumulative distribution function for the sum of the lengths of the two fuses. This is given by

=> F(y) = P(Y₁ + Y₂ ≤ y) = ∫∫f(x,y)dxdy,

where the integral is taken over the region x+y ≤ y. We can simplify this by changing the order of integration:

=> [tex]F(y) = \int0^y\int0^{y-x}f(x,y)dxdy.[/tex]

Using the probability density function given in part (a), we have

=> [tex]F(y) = \int0^y\int0^{y-x}(1/9)e^{-(x+y)/3}dxdy[/tex]

This can be solved using integration by parts or by using the fact that the exponential function integrates to itself, giving

=> [tex]F(y) = 1 - e^{-y/3)(y+3)}[/tex]

Finally, we can find P(Y₁ + Y₂ ≤ 1) by evaluating F(1) - F(0), which gives

=> [tex]P(Y_1 + Y_2 ≤ 1) = 1 - e^{(-1/3)(4/3)}[/tex].

This is a function that gives the probability that the total effective length of life of two fuses is less than or equal to one.

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Complete Question:

The length of life Y1 AND Y2 for fuses of a certain type is modeled by the exponential distribution, with

[tex]f(y) = \left \{ {1/3e^{-y/3} y > 0,} \atop {0, else where }} \right.[/tex]

(The measurements are in hundreds of hours.)

a) If two such fuses have independent lengths of life Y1 and Y2, find the joint probability density function for Y1 and Y2.

b) One fuse in part (a) is in a primary system, and the other is in a backup system that comes into use only if the primary system fails. The total effective length of life of the two fuses is then Y1 + Y2. Find P(Y1 + Y2 ≤ 1).