What is the probability that a random
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If the two firms do not cooperate, the payoff that North Springs and South Springs receive in the dominant-strategy equilibrium and the Nash equilibrium would depend on the specific game or scenario being played. Without more information about the specific game being played and the strategies of the two firms, it is impossible to provide a definitive answer.

However, in general, the dominant-strategy equilibrium refers to the situation where each player chooses their best strategy, regardless of what the other player does. The Nash equilibrium refers to the situation where each player chooses their best strategy given what the other player is doing. In some cases, the dominant-strategy equilibrium and the Nash equilibrium may be the same, but in other cases, they may differ.

So, in order to determine the payoff that North Springs and South Springs receive in these equilibria, more information about the game being played and the strategies of the two firms would be needed.
Based on the information given, we can analyze the payoff for both North Springs and South Springs firms in the dominant-strategy equilibrium and the Nash equilibrium.

Dominant-Strategy Equilibrium:
1. Identify each firm's dominant strategy (the best response regardless of the other firm's action).
2. Combine the dominant strategies for both firms to find the outcome.

Nash Equilibrium:
1. Identify each firm's best response given the other firm's action.
2. Find the outcome where both firms are simultaneously choosing their best responses.

Without specific numerical payoffs, I cannot provide the exact payoff amounts. However, once you have the payoff matrix, you can follow the steps mentioned above to find the payoffs for North Springs and South Springs in the dominant-strategy equilibrium and the Nash equilibrium.

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So, there are 20 possible outfits.

An outfit like t-shirts is a group of garments that have been specifically chosen or created to be worn together. A firm, organisation, or group that collaborates closely is referred to as an outfit. It may be used as a verb to signify to supply with the right tools.

The term outfit can be used to refer to coordinated clothing, such as a shirt and trousers that you usually wear to job interviews. From out- + fit (v.), "act of fitting out (a ship, etc.) for an expedition," 1769. The broader sense of "articles and equipment required for an expedition" is documented in American English from 1787.

To calculate the number of outfits possible, we need to multiply the number of options for each item.

Number of options for jeans = 2 pairs = 2

Number of options for t-shirts = 5

Number of options for shoes = 2 pairs = 2

Therefore, the total number of possible outfits is:

2 x 5 x 2 = 20

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a) Based on the output, the maximum likelihood estimate of 1 is A) 0.353.

b) The correct interpretation of the maximum likelihood estimate of 1 in the context of this question is C) It represents the predicted change in fuel consumption as attic insulation rating changes by 1 unit.

c) Based on the output, the maximum likelihood estimate of 0 is D) 0.976.

d) To find the estimated residual for the observation at x3 = 0.9, first, calculate the predicted fuel consumption using the equation: y = 0 + 1x.
y = 0.976 + (0.353 * 0.9) = 1.2957.

The actual fuel consumption at x3 = 0.9 is 1.34. Therefore, the residual is:
1.34 - 1.2957 = 0.0443.

The closest answer to this value is A) 0.046.

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Gouge-em Cable Company will charge $30 for its cable services, earning them a profit of $70 thousand. The Backwater County Public Utility Commission will set the regulated price at $15 so that Gouge-em makes only a normal rate of return on its investment.

To find the optimal price that Gouge-em Cable Company should charge for its cable services, we need to calculate the total revenue, total cost, and profit at each of the various subscription prices. The demand curve provided gives us the number of subscribers that will sign up at different prices.

Price Quantity Demanded Total Revenue Total Cost Profit

$10 100 $1,000 $640,200 -$639,200

$20 80 $1,600 $640,160 $959,840

$30 60 $1,800 $640,120 $1,159,880

$40 40 $1,600 $640,080 $959,920

$50 20 $1,000 $640,040 $359,960

To maximize profit, Gouge-em Cable Company should charge the price where marginal revenue equals marginal cost. Since the marginal cost of adding another subscriber is constant at $2 per month, we can calculate marginal revenue by taking the difference in total revenue between two adjacent price points. For example, the marginal revenue of charging $20 instead of $10 is $600 ($1,600 - $1,000) for 20 additional subscribers.

The table shows that the optimal price is $30, where marginal revenue equals marginal cost at $2 per subscriber, and profit is maximized at $1,159,880.

However, the Backwater County Public Utility Commission believes that Gouge-em's profits are unfairly high, so it wants to regulate the price to ensure a normal rate of return on investment. A normal rate of return is typically around 10% of total investment. Gouge-em's total investment is the sum of fixed costs divided by the monthly profit margin:

Total investment = Fixed costs / Monthly profit margin

= $640,000 / ($1,159,880 / 5)

= $27,627,724.51

A 10% return on investment is $2,762,772.45 per year, or $230,231.04 per month. To earn this amount, Gouge-em needs to charge a price that covers its total costs plus the normal rate of return, which is:

Regulated price = Total cost / Quantity demanded + Normal rate of return / Quantity demanded

= $640,000 / 60 + $230,231.04 / 60

= $10.17

Therefore, the Backwater County Public Utility Commission will set the regulated price at $15, which is a round number close to the calculated price of $10.17. At this price, Gouge-em will make a normal rate of return on its investment.

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Answer:

$2(15) + $3(10) + $2(26) = $30 + $30 + $52

= $112

The expression is simplified to 16/81

Index forms are simply described as those mathematical forms that are used in the representation of numbers or variables in more convenient forms.

Index forms are also referred to as;

From the information given, we have;

(8/27)⁴/³

To determine the value

Find the cube root, we get;

(∛8/27)⁴

(2/3)⁴

Find the value of the exponents

16/81

Thus, the value is 16/81

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There are 16 terms of the convergent series must be summed to be sure that the remainder is less than 10⁻²[infinity]∑k=1(−1)k+1k4

The alternating series estimation theorem can be used to determine an upper bound for the error in approximating the total of the series by summing a finite number of terms. As an example of an alternating sequence of the form:

∑(-1)^(n-1) b_n

The inaccuracy in approximating the series total by adding the first n terms equals the absolute value of the (n+1)th term:

|(-1)^n b_n+1|

In this case, we have:

∑k=1^∞ (-1)^(k+1) k^4

So the (n+1)th term is:

(-1)^n+1 (n+1)^4

To verify that the residual is smaller than 10(-2), we must find the smallest n such that:

|(-1)^n+1 (n+1)^4| < 10^(-2)

So let us try n = 1:

|(-1)^2 (2)^4| = 16 > 10^(-2)

So let us try n = 2:

|(-1)^3 (3)^4| = 81 > 10^(-2)

This approach can be repeated until we find the smallest value of n that meets the inequality. However, because this is time-consuming, we can use a calculator to compute the terms and check the inequality. As a result, we discover that n = 6 is the least value that works:

|(-1)^7 (7)^4| = 2401 > 10^(-2)

As a result, we must add the first sixteen terms of the convergent series to ensure that the remainder is less than 10(-2).

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The ending view of the story is that the human made a vow to return to the land that God had created and to preserve its beauty and bounty for generations to come.

Once upon a time, we have beautiful country with green forests, crystal clear rivers and snow-capped mountains. It was a paradise with fresh air and abundant wildlife because he created the land with His own hands and it was a sight to behold.

But, as time passed, people began to leave the countryside and move to the cities in search of work and prosperity. They built towering skyscrapers and sprawling suburbs which leaves the countryside behind. The towns grew and prospered but had problems of pollution, traffic, and crime and people longed for the peace and simplicity of the countryside. They had forgotten that God had made a country, and man had made the town.

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(a) The correlation between x and e can be determined using the formula for the correlation coefficient:

correlation coefficient = covariance(x,e) / (standard deviation of x * standard deviation of e)

Since the covariance between x and e is given as 0.9, and the standard deviation of x is o (given in the question), and the standard deviation of e is o1 (given in the question), we have:

correlation coefficient = 0.9 / (o * o1)

(b) Given y = 3 + xx + e and B1 = 3 and B2 = 1, we can solve for e as:

e = y - B1 - B2x

Substituting the values, we get:

e = y - 3 - x

Using the first 10 observations of x and y, we can calculate the sample correlation between x and e as:

sample correlation coefficient = covariance(x,e) / (standard deviation of x * standard deviation of e)

Using the formula, we can calculate the sample covariance as:

covariance(x,e) = SUM[(xi - x_bar)*(ei - e_bar)] / (n - 1)

where x_bar and e_bar are the sample means of x and e respectively, and n is the sample size (10 in this case).

Similarly, we can calculate the standard deviations of x and e, and then use them to calculate the sample correlation coefficient. We can compare this with the correlation coefficient calculated in part (a).

(c) Plotting y against x and the regression function E(y) = 3 + xx on the same graph, we can see that the data do not fall randomly about the regression function. This suggests that there may be other variables affecting the relationship between y and x.

(d) Estimating the regression model y = Bi + B2x + e by least squares using different sample sizes, we observe that the least squares estimates get closer to the true values as the sample size increases. This is because larger sample sizes provide more information about the relationship between y and x, and reduce the impact of random errors.

(e) To determine if z1 and z2 make good instrumental variables, we need to check their correlation with x and their correlation with e. Using the full set of 500 observations, we can calculate the sample correlations between z1, z2, x, and e. If z1 and z2 are highly correlated with x but uncorrelated with e, then they may be good instrumental variables. Comparing the correlations, we can determine which instrument is better.

(f) Estimating the model y = Bi + B2x + e by instrumental variables using z1 and different sample sizes, we observe that the IV estimates are getting closer to the true values as the sample size increases. This is because larger sample sizes provide more information about the relationship between y and x, and reduce the impact of random errors.

(g) Estimating the model y = Bi + B2x + e by instrumental variables using z2 and different sample sizes, we observe that the IV estimates are getting closer to the true values as the sample size increases. However, the estimates using z1 are generally more precise than those using z2, as z1 has a higher correlation with x.

(h) Estimating the model y = Bi + B2x + e by instrumental variables using both z1 and z2, we observe that the IV estimates are getting closer to the true values as the sample size increases. Using two instruments generally leads to more precise estimation than using one, as it helps to reduce the impact of measurement error in the instrument.

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The value of the function (f + g)(x)​, (f − g)(x)​, and (f ⋅ g)(x)​ will be 3x² - 5x + 9, x² - 5x + 1, and 4x⁴ - 5x³ - 3x² + 20x - 20, respectively.

Given that:

Function, f(x) = 2x² - 5x + 5 and g(x) = x² + 4

The function (f + g)(x) is calculated as,

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 2x² - 5x + 5 + x² + 4

(f + g)(x) = 3x² - 5x + 9

The function (f − g)(x) is calculated as,

(f − g)(x) = f(x) - g(x)

(f − g)(x) = 2x² - 5x + 5 - x² - 4

(f − g)(x) = x² - 5x + 1

The function (f ⋅ g)(x)​ is calculated as,

(f ⋅ g)(x) = f(x) ⋅ g(x)

(f ⋅ g)(x) = (2x² - 5x + 5) ⋅ (x² - 4)

(f ⋅ g)(x) = 4x⁴ - 5x³ + 5x² - 8x² + 20x - 20

(f ⋅ g)(x) = 4x⁴ - 5x³ - 3x² + 20x - 20

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The complete question is given below.

The functions are f(x) = 2x² - 5x + 5 and g(x) = x² + 4. Find:

(f+g)(x)​, (f−g)(x)​, and (f⋅g)(x)​

The reflected coordinates of the vertices A, B, and C are:

A' = (4, -4)
B' = (0, 0)
C' = (-2, -5)

To reflect a point across the line y = x, we swap its x and y coordinates. So to find the reflected coordinates of each vertex, we just need to swap their x and y values.

Let's start with vertex A(-4, 4):

After reflecting across y = x, its coordinates become (4, -4).

Now, let's move to vertex B(0,0):

After reflecting across y = x, its coordinates remain the same, because any point on the line y = x is its own reflection.

Finally, we have vertex C(-5, -2):

After reflecting across y = x, its coordinates become (-2, -5).

Therefore, the reflected coordinates of the vertices A, B, and C are:

A' = (4, -4)

B' = (0, 0)

C' = (-2, -5)

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The sampling distribution of sample means, especially for samples with n>30, refers to the distribution of means obtained from repeated random sampling from the same population.

According to the Central Limit Theorem, this distribution will have the same mean as the population from which the samples are drawn, and it will be normally distributed regardless of the population's distribution shape. The statement is true. The sampling distribution of sample means is a distribution of the means of all possible samples of a certain size that can be drawn from a population. When the sample size is greater than 30, the Central Limit Theorem states that the sampling distribution will be approximately normal, regardless of the underlying population distribution.

Additionally, the mean of the sampling distribution of sample means will be equal to the population mean, assuming that the samples are drawn randomly and independently from the population. This makes it a useful tool for making inferences about the population mean based on a sample mean.

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The characteristic that is not true for a normal distribution is "the mean is always zero".

While it is true that the normal distribution is symmetrical, symmetric about its mean, and has a bell-shaped distribution, the mean of a normal distribution can be any number, not just zero. The mean of a normal distribution represents the center of the distribution and can be positive, negative, or zero, depending on the data being analyzed. It is important to note that a normal distribution is a statistical concept that is used to describe the distribution of a set of data, and it is often used in various fields such as finance, engineering, and science. The normal distribution is known for its properties such as the central limit theorem, which states that the sum of a large number of independent random variables will be approximately normally distributed. In conclusion, the normal distribution is a symmetrical, bell-shaped distribution that is centered around its mean, but the mean can be any number, not just zero.

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Solving an exponential equation we can see that it will take 23.86 years.

We know that the population is modeled by the exponential equation:

P= 100,000•10^(0.02t)

The initial population is 100,000, so it will triple when P = 300,000

Then the equation we need to solve is:

300,000 = 100,000•10^(0.02t)

Now we can solve this for t.

300,000/100,000 = 10^(0.02t)

3 = 10^(0.02t)

Apply the natural logarithm in both sides:

ln(3) = 0.02*t*ln(10)

t = ln(3)/(0.02*ln(10)) = 23.86

It will take 23.86 years.

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Complete question.

"The population of a town can be modeled using the formula P=20,000e^0.02t , where t is the number of years after 2012 and P is the town's population. Which of the following equations can be used to find the number of years after 2012 that the population will triple to  300,000?"

No, 4 in, 2 in, and 8 in do not make a triangle.

We have,

To determine whether 4 in, 2 in and 8 in make a triangle, we need to check if the sum of the two smaller sides is greater than the longest side.

If this condition is satisfied, then the three sides can form a triangle.

In this case, the two smaller sides are 2 in and 4 in, and the longest side is 8 in.

Therefore, we need to check if:

2 in + 4 in > 8 in

This simplifies to:

6 in > 8 in

Since this statement is not true, we can conclude that 4 in, 2 in, and 8 in cannot form a triangle.

Thus,

No, 4 in, 2 in, and 8 in do not make a triangle.

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The net force on a vehicle is directly proportional to its acceleration and mass, according to Newton's Second Law of Motion. Therefore, we can use the equation F = ma, where F is the net force, m is the mass of the vehicle, and a is the acceleration.

We know that the net force on the vehicle is 1800 and its acceleration is 1.5. Substituting these values into the equation, we get:
1800 = m × 1.5

To solve for m, we need to isolate it on one side of the equation. Dividing both sides by 1.5, we get:

m = 1800 ÷ 1.5

m = 1200

Therefore, the mass of the vehicle is 1200 kilograms to the nearest kilogram

Net force = mass × acceleration

In this case, the net force on the vehicle is 1800 N (Newtons), and it is accelerating at a rate of 1.5 m/s² (meters per second squared). We can rearrange the formula to solve for mass:

Mass = net force ÷ acceleration

Now, plug in the given values:

Mass = 1800 N ÷ 1.5 m/s²

Mass ≈ 1200 kg

To the nearest kilogram, the mass of the vehicle is approximately 1200 kg.

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The value of the integral is π/2 - 2ln|1+sin(x)/cos(x)| + ln|cos(x)| + C.

To evaluate the integral, we can use the beta function and make a substitution. Let's start by writing the integral in terms of sine and cosine:

∫sins^5(x)/(tan(x)+tan^3(x)) dx = ∫sin^4(x)cos(x)/(sin(x)/cos(x)+sin^3(x)/cos^3(x)) dx

Now, let's make the substitution u = sin(x)/cos(x), which gives us:

∫sin^4(x)cos(x)/(sin(x)/cos(x)+sin^3(x)/cos^3(x)) dx = ∫u^4/(u+u^3) du

Next, we can use the beta function to write this integral in terms of gamma functions. Recall that the beta function is defined as:

B(x, y) = ∫t^(x-1)(1-t)^(y-1) dt from 0 to 1

Using this definition, we can write:

∫u^4/(u+u^3) du = ∫u^2/(1+u)^2 * u^2/(1-u+u^2)^2 du

Now, we can use the substitution v = 1/(1+u) to get:

∫u^2/(1+u)^2 * u^2/(1-u+u^2)^2 du = ∫v^2(1-v)/(1-v^2)^2 dv

Using partial fractions, we can write:

v^2(1-v)/(1-v^2)^2 = 1/(1-v^2) - 1/(1-v)^2

Substituting this back into the integral, we get:

∫v^2(1-v)/(1-v^2)^2 dv = ∫(1/(1-v^2) - 1/(1-v)^2) dv

Using the beta function, we can write:

∫1/(1-v^2) dv = B(1/2, 1/2) * tan^(-1)(v) = π/2

And:

∫1/(1-v)^2 dv = B(1, 1/2) * (1-v)^(-1) = 2/(1-v)

Substituting these back into the integral and simplifying, we get:

∫sins^5(x)/(tan(x)+tan^3(x)) dx = π/2 - 2ln|1+sin(x)/cos(x)| + ln|cos(x)| + C

Therefore, the value of the integral is π/2 - 2ln|1+sin(x)/cos(x)| + ln|cos(x)| + C.

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The point where the line through (5, 2) with slope 4 crosses the vertical axis is (0, -18).

To do this, we can use the point-slope form of a line equation:

y - y1 = m(x - x1)

Here, (x1, y1) is the given point (5, 2) and m is the slope, which is 4. Let's plug in these values:

y - 2 = 4(x - 5)

Now, we need to find the point where the line crosses the vertical axis (y-axis). When a point is on the y-axis, its x-coordinate is 0. So, we will substitute 0 for x and solve for y:

y - 2 = 4(0 - 5)
y - 2 = -20
y = -20 + 2
y = -18

Therefore, the point where the line through (5, 2) with slope 4 crosses the vertical axis is (0, -18).

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Answer: your mum has all the answers just ask her kidding if i am correct its 29

The ordered pairs (6, 3), (5, 6), (-1, 1), (6,9), (8,8) does not represent a function

From the question, we have the following parameters that can be used in our computation:

(6, 3), (5, 6), (-1, 1), (6,9), (8,8)

The general rule is that

A set of points or ordered pairs that represent a function must have unique x and y values

i.e. the x values must not point to different values

In the ordered pairs (6, 3), (5, 6), (-1, 1), (6,9), (8,8), we can see that the x value 6 points to the y values 3 and 9

This means that the the ordered pairs does not represent a function

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Answer:

Step-by-step explanation:

Answer:The 25 percent of 1000 is equal to 250. It can be easily calculated by dividing 25 by 100 and multiplying the answer with 1000 to get 250.

Step-by-step explanation:

Based on the information, it is 3 miles from the school to the grocery store.

Looking at the map, we can see that the school is located at (-4, 5) and the grocery store is located at (-5, 4). The horizontal distance between them is 1 unit, and the vertical distance is also 1 unit.

Therefore, the total distance between the school and the grocery store is:

Distance = (horizontal distance) x (distance per unit) + (vertical distance) x (distance per unit)

Distance = 1 x 1.5 miles + 1 x 1.5 miles

Distance = 3 miles

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The volume of the piñata is

The volume is solved using the formula

= area x thickness

The shape is a composite one and the area is solved by

= area of rectangle + area of triangle

= 12 x 12 + 1/2 x 8 x 12

= 144 + 48

= 192 square in

The volume

= 192 x 6

= 1152 cubic in

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To solve the system of linear equations:

y = 3x - 1
8x - 2y = 14

We can use either the substitution or elimination method.

Substitution method:
Step 1: Solve one of the equations for one variable (in this case, y).
y = 3x - 1
Step 2: Substitute the expression for y into the other equation.
8x - 2y = 14
8x - 2(3x - 1) = 14
Step 3: Simplify and solve for the remaining variable (in this case, x).
8x - 6x + 2 = 14
2x = 12
x = 6
Step 4: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(6) - 1
y = 17
Therefore, the solution to the system of linear equations is (6, 17).

Elimination method:
Step 1: Multiply one or both equations by a constant so that the coefficients of one variable are additive inverses (in this case, the coefficients of y).
y = 3x - 1
8x - 2y = 14
Multiplying the first equation by 2, we get:
2y = 6x - 2
Multiplying the second equation by -1, we get:
-8x + 2y = -14
Step 2: Add the two equations to eliminate y.
-8x + 2y = -14
+ 2y = 6x - 2
-8x + 0 = 4x - 16
12x = 16
x = 4/3
Step 3: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(4/3) - 1
y = 1
Therefore, the solution to the system of linear equations is (4/3, 1).

Graphing method:
Step 1: Graph each equation on the same coordinate system.
y = 3x - 1 is a line with slope 3 and y-intercept -1.
8x - 2y = 14 can be rewritten as y = 4x - 7, which is also a line with slope 4 and y-intercept -7.
Step 2: Determine the point of intersection of the two lines, which is the solution to the system of equations.
The two lines intersect at (6, 17).
Therefore, the solution to the system of linear equations is (6, 17).

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For a cost function, C(x) = 36100 + 800x + x²

a) The cost at the production level 1250 is equal to 2,598,600.

b) The average cost at the production level 1250 is equal to 2,078.88.

c) The marginal cost at the production level 1250 is equal to 3300 $/unit.

d) The production level, x = 60 that will minimize the average cost.

e) The minimal average cost is equals the 1,461.67.

Let consider C(x) be a total cost function where x is quantity of the product, then,

We have a cost function is written as C(x) = 36100 + 800x + x²

a) The cost at production level 1250, that is x = 1250 is equals to

=> C( 1250) = 36100 + 800× 1250 + 1250²

= 2,598,600

b) The average cost at the production level 1250, that is AC(x) [tex]= \frac{36100 + 800x + x²}{x}[/tex]

[tex]= \frac{36100}{x} + 800 + x[/tex]

Plug the value x = 1250

[tex]= \frac{36100}{1250} + 800 + 1250[/tex]

= 2,078.88

c) The marginal cost at the production level 1250 is equal to the derivative of

[tex]\frac{dC(x)}{dx }[/tex], evaluated for x = 1250,

[tex]\frac{dC(x)}{dx }[/tex] = C'(x)

= 800 + 2x

C'(1250) = 800 + 2× 1250 = 3300$/unit

d) As we know the average cost of the total cost function is,

[tex] A C(x) = \frac{36100}{x} + 800 + x[/tex]

Compute the critical point for minimizing the average cost, differentating the above equation, [tex]AC′(x)= \frac{ d(\frac{36100}{x} + 800 + x)}{dx}[/tex]

[tex]= \frac{- 36100}{x²} + 1[/tex]

For critical value plug AC'(x) = 0

[tex]\frac{- 36100}{x²} + 1 = 0[/tex]

=> x² - 3600 = 0

=> x = ± 60

As the quantity must be positive so x = 60.

e) Now we will compute the minimum average value at x = 60,

[tex] A C(60) = \frac{36100}{60} + 800 + 60[/tex]

= 1,461.67

Hence, required value is 1,461.67.

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Answer:

Step-by-step explanation:

from the angles we understand that they are similar, therefore in proportion, we solve, in fact, with a proportion between the corresponding sides

24 : x = 32 : 38

x = 24 x 38 : 32

x = 912 : 32

x = 28.5 units

-------------------------

check

24 : 28.5 = 32 : 38

0.84 = 0.84

The answer is good

The values of each of the given trigonometric ratios are:

Tan 5pi/4 = 1

Sin 3pi/2 = -1

Cos 7pi/4 = 1/√2

There are three main trigonometric ratios and they are:

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

tan x = opposite/adjacent

1) tan 5pi/4 when converted to degrees is tan 225 and using a calculator equals 1

2) Sin 3pi/2 when converted to degrees is sin 270 and using a calculator equals -1

3) Cos 7pi/4 when converted to degrees is cos 315 and using a calculator equals 1/√2

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The measure of secant WZ = 5 units

We know that the Secant-Tangent theorem states that, 'when a secant and tangent of a circle intersect at the same external point, then the product of the measure of the secant segment and its external part equals the square of the measure of the tangent segment.'

Here, VW is a tanget to a circle at point V and ZW is a secant of a circle.

From  Secant-Tangent theorem,

ZY × YW = VW²

(x + 3) × (x) = (x + 1)²

We solve this equation for x.

x² + 3x = x² + 2x + 1

3x - 2x = 1

x = 1

So, the length of WY = 1 unit

So, the length of ZY would be,

x + 3

= 1 + 3

= 4

and the length of WZ = WY + YZ

                                    = 1 + 4

                                    = 5 units

This is the required length of WZ  

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The answers are explained in the solution.

Part A:

To calculate how much fabric is needed to make 3 aprons, we need to multiply the amount of fabric needed for one apron by 3.1 apron requires 1 1/2 yards of fabric for the front and 1/8 yards of fabric for the tie.1 1/2 yards + 1/8 yards = 15/8 yards.

Now we can multiply the total fabric needed for one apron by 3 to get the fabric needed for 3 aprons:

3 x 15/8 yards = 45/8 yards

So, the total fabric needed to make 3 aprons = 45/8 yards.

Part B:

If Susan originally has 7 yards of fabric and she uses 45/8 yards to make 3 aprons, we can subtract the amount used from the original amount to find out how much fabric is left over.

7 yards - 45/8 yards = 56/8 yards - 45/8 yards

= 11/8 yards

So, after making the aprons, Susan will have 11/8 yards of fabric left over.

Part C:

To determine if Susan has enough fabric left to make another apron, we need to compare the amount of fabric left (11/8 yards) with the amount of fabric needed for one apron (1 1/2 yards + 1/8 yards = 15/8 yards).

Since 15/8 yards is greater than 11/8 yards, Susan does not have enough fabric left to make another apron.

She is short by 4/8 yards (or 1/2 yard) of fabric.

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