The quality control section of an industrial firm uses systematic sampling to estimate the average amount of fill in 12-ounce cans coming off an assembly line. The data in the accompanying table represent a 1-in-50 systematic sample of the production in one day. Estimate m and place a bound on the error of estimation. Assume N = 1800.

Answer:

x = 5. The number added to the numerator and to the denominator is 5.

Step-by-step explanation:

Lets start with 3/5. We're going to add the same number to the top and the bottom. We don't know that number, so we use x.

(3+x)/(5+x)

They said that then becomes 4/5.

(3+x)/(5+x) = 4/5

crossmultiply.

5(3+x) = 4(5+x)

use distributive property.

15 + 5x = 20 + 4x

subtract 4x from both sides.

15 + x = 20

subtract 15 from both sides.

x = 5

Check:

(3+5)/(5+5)

= 8/10

= 4/5

Carlos needs 636 square feet of insulation to properly insulate the room he just finished framing in his home.

First, we need to calculate the square footage of the walls by multiplying the perimeter of the room by the height of the walls. The perimeter of the room is calculated by adding up the length of all four walls. In this case, the perimeter is 2(16 ft.) + 2(12 ft.) = 56 ft. Therefore, the total square footage of the walls is 56 ft. x 9 ft. = 504 ft².

Next, we need to calculate the square footage of the ceiling. The ceiling measures 16 ft. by 12 ft., so the total square footage is 16 ft. x 12 ft. = 192 ft².

Finally, we need to account for the windows in the room. The total square footage of the windows is 2(5 ft. x 6 ft.) = 60 ft².

To determine the total square footage of insulation needed, we add the square footage of the walls and ceiling and subtract the square footage of the windows. Therefore, the total square footage of insulation needed is (504 ft² + 192 ft²) - 60 ft² = 636 ft².

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The exact value of the integral is approximately 18.184, while the trapezoidal rule approximation with n=5 is approximately 18.178.

To apply the trapezoidal rule, we need to divide the interval [6,10] into n=5 subintervals of equal width:

Δx = (10-6)/5 = 1.6

The endpoints of these subintervals are:

x0 = 6

x1 = 6 + Δx = 7.6

x2 = 6 + 2Δx = 9.2

x3 = 6 + 3Δx = 10.8

x4 = 6 + 4Δx = 12.4

The trapezoidal rule states that:

[tex]\int _a^b f(x) dx \approx \Delta x/2 [f(a) + 2f(x1) + 2f(x2) + ... + 2f(x(n-1)) + f(b)][/tex]

Applying this formula with a=6, b=10 and n=5, we have:

[tex]10\int 6^{10} 3/(x-4) dx \approx x/2 [f(6) + 2f(7.6) + 2f(9.2) + 2f(10.8) + f(12.4)][/tex]

where f(x) = 3/(x - 4)

f(6) = 3/(6-4) = 1.5

f(7.6) = 3/(7.6-4) = 0.7299

f(9.2) = 3/(9.2-4) = 0.5

f(10.8) = 3/(10.8-4) = 0.375

f(12.4) = 3/(12.4-4) = 0.2909

Substituting these values, we get:

[tex]10\int 6^{ 10} 3/(x-4) dx \approx 0.8 [1.5 + 2(0.7299) + 2(0.5) + 2(0.375) + 0.2909][/tex]

[tex]10\int 6^{10} 3/(x-4) dx \approx 18.178[/tex]

To find the exact value of the integral, we can use the antiderivative of f(x):

∫ 3/(x-4) dx = 3 ln|x-4| + C

where C is the constant of integration.

Using this formula, we have:

[tex]10\int 6^{ 10} 3/(x-4) dx = [10(3 ln|x-4|)]_ 6^10[/tex]

= 30 ln|10-4| - 30 ln|6-4|

= 30 ln(3) - 30 ln(2)

≈ 18.184.

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a. The PDF of Y is:

[tex]f_Y(y) = d/dy F_Y(y) = 3(1-y)^2,[/tex] 0 <= y <= 1.

b. The PDF of Z is:

[tex]f_Z(z) = d/dz F_Z(z) = 3z^2, 0 < = z < = 1.[/tex]

To find the PDF of Y and Z, we need to use the following properties of uniform random variables:

The PDF of a uniform random variable [tex]U~Unif[a, b][/tex] is f(u) = 1/(b-a) for a <= u <= b, and 0 otherwise.

If U1, U2, ..., Un are independent uniform random variables, then the joint PDF of (U1, U2, ..., Un) is f(u1, u2, ..., un) = 1/(b-a)^n for a <= ui <= b, i = 1, 2, ..., n, and 0 otherwise.

a. To find the PDF of Y = max{X1, X2, X3}, we first need to find the CDF of Y:

[tex]F_Y(y) = P(Y < = y) = 1 - P(Y > y)[/tex] = 1 - P(X1 > y, X2 > y, X3 > y)

= 1 - P(X1 > y)P(X2 > y)P(X3 > y) (by independence)

[tex]= 1 - (1-y)^3 (since X~Unif[0,1] and P(X > x)[/tex] = 1 - P(X <= x) = 1 - x)

So, the PDF of Y is:

[tex]f_Y(y) = d/dy F_Y(y) = 3(1-y)^2,[/tex] 0 <= y <= 1.

b. To find the PDF of Z = min{X1, X2, X3}, we need to find the CDF of Z:

[tex]F_Z(z)[/tex]= P(Z <= z) = P(X1 <= z, X2 <= z, X3 <= z)

= P(X1 <= z)P(X2 <= z)P(X3 <= z) (by independence)

[tex]= z^3.[/tex]

So, the PDF of Z is:

[tex]f_Z(z) = d/dz F_Z(z) = 3z^2, 0 < = z < = 1.[/tex]

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The value of the test statistic for testing H0: rho = 0 vs. H1: rho ≠ 0, given a correlation coefficient of 0.47 for a sample of 17 students, is approximately 2.06.

To find the value of the test statistic for testing H0: rho = 0 vs. H1: rho ≠ 0, given a correlation coefficient of 0.47 for a sample of 17 students, you can follow these steps:

Step 1: Calculate the degrees of freedom.
Degrees of freedom (df) = n - 2, where n is the sample size.
df = 17 - 2 = 15

Step 2: Use the formula for the test statistic, t:
t = (r * sqrt(df)) / sqrt(1 - r^2), where r is the correlation coefficient.
t = (0.47 * sqrt(15)) / sqrt(1 - 0.47^2)

Step 3: Calculate the value of the test statistic:
t = (0.47 * sqrt(15)) / sqrt(1 - 0.2209)
t = (0.47 * 3.87298) / sqrt(0.7791)
t = 1.8202046 / 0.88270386

t ≈ 2.06

Thus, The value of the test statistic is approximately 2.06.

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Lateral Surface Area of triangle prism = (2a + c) x h

We need to determine the area of all the rectangular sides of a triangular prism in order to calculate its lateral surface area. In this instance, there are two isosceles triangles and one rectangle.

The following formula can be used to determine the triangular prism's lateral surface area:

Lateral Surface Area = Perimeter of Base x Height

The sum of the lengths of each side of a triangular prism forms the base's perimeter.

Since the base is an isosceles triangle in this case, the perimeter can be calculated by dividing the length of the third side by twice the length of one of the equal sides.

We should expect that the foundation of the three-sided crystal has sides of length a, b, and c. We can expect to be that an and b are the equivalent sides, and c is the third side.

Perimeter of Base = 2a + c

The height of the triangular prism is the perpendicular distance between the two parallel bases, which is the length of the rectangular face of the prism. Let's assume that the height of the triangular prism is h.

Now, the lateral surface area of the triangular prism can be found by multiplying the perimeter of the base by the height of the prism:

Lateral Surface Area = Perimeter of Base x Height

Lateral Surface Area = (2a + c) x h

Therefore, to calculate the lateral surface area of the triangular prism, we need to know the length of the sides of the base, c and the height of the prism, h. Once we have these values, we can use the formula to find the lateral surface area.

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Hello! Today, we'll be discussing the Ratio Test and how it can be used to find the values of x for which the series [infinity]Σ (x-6)² / k² converges. This is a common problem in calculus and can be approached in a few different ways, but we'll focus on the Ratio Test method.

The series converges for x in the interval (5, 7), or equivalently, x ∈ (5, 7).

The Ratio Test is a powerful tool used to determine the convergence or divergence of an infinite series.

We want to determine the values of x for which this series converges, so we'll apply the Ratio Test by taking the ratio of consecutive terms:

|(x-6)² / (k+1)²| / |(x-6)² / k²|

We can simplify this expression by multiplying both the numerator and denominator by k² and cancelling out the (x-6)² terms:

|k² / (k+1)²|

To evaluate this limit, we can use L'Hopital's rule or simply expand the denominator and simplify:

k² / (k+1)² = k² / (k² + 2k + 1) = 1 / (1 + 2/k + 1/k²)

As k approaches infinity, the terms 2/k and 1/k² both approach zero, so the limit simplifies to 1. Therefore, the ratio of consecutive terms approaches 1 as k approaches infinity, which means the Ratio Test is inconclusive.

However, we can still determine the values of x for which the series converges by looking at the behavior of the series for specific values of x.

Since the numerator is always zero, this series converges by the Comparison Test. Similarly, if we let x = 5 or x = 7, then the series diverges by the Comparison Test.

To find the full range of values of x for which the series converges, we can use the fact that the series converges for x = 6 and diverges for x < 5 and x > 7.

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The variance for the sample mean is 0.0001 cm².

To find the variance for the sample mean of 4 computer chips with a mean of 0.95 centimeters and a standard deviation of 0.02 centimeters, you can follow these steps:

1. Note the population mean (µ) = 0.95 cm and population standard deviation (σ) = 0.02 cm.
2. Identify the sample size (n) = 4.
3. Calculate the variance for the sample mean using the formula: variance of sample mean = (σ²) / n.

In this case, the variance of the sample mean is:

Variance of sample mean = (0.02 cm)² / 4 = 0.0004 cm² / 4 = 0.0001 cm².

So, the variance for the sample mean of the 4 computer chips is 0.0001 cm².

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the manager’s choice of 5 percent as the target service level is justified based on the Newsvendor model

The manager’s choice of 5 percent is a reasonable decision as it ensures that the percentage of customers facing a shortage is kept at a low level, which is important for maintaining customer satisfaction and long-term profits.

To understand the impact of this decision on the ordering policy, we can use the Newsvendor model, which is a widely used model in inventory management. The Newsvendor model calculates the optimal order quantity that minimizes the expected cost of ordering too much or too little inventory.

Using the given information, the cost of ordering too much inventory is the setup cost of $80, while the cost of ordering too little inventory is the shortage cost of $100. The expected profit per refrigerator sold is $100 ($1300 - $1200) and the inventory carrying rate is 35% of the cost of the refrigerator, which is $420 ($1200 x 35%).

The optimal order quantity, Q, can be calculated as follows:

Q = mean demand during lead time + z * standard deviation of demand during lead time

where z is the z-score associated with the service level, which is the complement of the percentage of customers facing a shortage. For a service level of 95%, the z-score is 1.645.

Using the given mean and standard deviation of demand, and the mean and standard deviation of lead time, we can calculate the expected demand during lead time and the standard deviation of demand during lead time as follows:

Expected demand during lead time = mean demand x mean lead time = 5 x 5 = 25

Standard deviation of demand during lead time = standard deviation of demand x square root of lead time = 2 x sqrt(5) = 4.47

Substituting these values into the Newsvendor formula, we get:

Q = 25 + 1.645 x 4.47 = 32.38

Rounding up to the nearest integer, the optimal order quantity is 33 refrigerators.

To check if this order quantity meets the manager’s service level target of 95%, we can calculate the actual service level using the cumulative distribution function (CDF) of the Normal distribution as follows:

Actual service level = 1 - CDF(z-score) = 1 - CDF(1.645) = 1 - 0.9505 = 0.0495

This means that the percentage of customers facing a shortage is approximately 5%, which meets the manager’s target.

Therefore, the manager’s choice of 5 percent as the target service level is justified based on the Newsvendor model

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There are 1820 different committees that can be chosen from the 16-member student council.

What is probability?

Probability is a measure of the likelihood of an event occurring.

The number of different committees that can be chosen from a group of n members, when choosing k members at a time, is given by the combination formula:

C(n, k) = n! / (k! * (n - k)!)

In this case, there are 16 students in the council and we need to choose a committee of 4 students. So we can substitute n=16 and k=4 into the formula to get:

C(16, 4) = 16! / (4! * (16 - 4)!) = 16! / (4! * 12!) = (16 * 15 * 14 * 13) / (4 * 3 * 2 * 1) = 1820

Therefore, there are 1820 different committees that can be chosen from the 16-member student council.

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The mean of the number of houses burglarized in this sample is 4.5 based on probability.

The probability of a house in an urban area being burglarized is 15%. This means that out of every 100 houses, 15 will be burglarized.

If 30 houses are randomly selected, we can use the binomial distribution to find the mean number of houses burglarized. The formula for the mean of a binomial distribution is:

Mean = n * p

where n is the number of trials (in this case, 30) and p is the probability of success (in this case, 0.15).

Substituting these values, we get:

Mean = 30 * 0.15
Mean = 4.5

Therefore, the mean number of houses burglarized out of 30 randomly selected houses is 4.5. Note that this is an expected value and may not represent the actual number of houses burglarized in any particular sample of 30 houses.
To find the mean of the number of houses burglarized, we can use the formula:

Mean = (Probability of success) x (Number of trials)

In this case:
- Probability of success (a house being burglarized) is 15%, or 0.15 as a decimal.
- Number of trials (randomly selected houses) is 30.

Using the formula, we get:

Mean = (0.15) x (30) = 4.5

So, the mean of the number of houses burglarized in this sample is 4.5.

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The probability of getting exactly 7 calls within the next three hours is approximately 0.090. C

The number of calls follows a Poisson distribution with a mean of 5 calls per hour, we can use the Poisson probability formula to find the probability of getting exactly 7 calls in 3 hours:

[tex]P(X = 7) = (e^{(-\lambda)} \times \lambda^x) / x![/tex]

λ is the mean rate of calls per hour, and x is the number of calls we are interested in over a duration of 3 hours.

In this case,[tex]\lambda = 5[/tex] calls per hour and x = 7 calls in 3 hours. So we have:

[tex]P(X = 7) = (e^{(-53)} \times (53)^7) / 7![/tex]

[tex]P(X = 7) \approx 0.090[/tex]

The probability of getting exactly 7 calls within the next three hours is approximately 0.090.

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The evaluated integral is 4.

To evaluate the integral ſf (4x + (4x + 2)dA where D is the region bounded by the curves y = x and y = 2x, we first need to set up the limits of integration.

Since the region is bounded by y = x and y = 2x, we know that the x limits are from x = 0 to x = 1 (where the two curves intersect).

Next, we need to find the y limits for each value of x. For a given value of x, the lower y limit is y = x, and the upper y limit is y = 2x.

Therefore, the integral becomes:
ſf (4x + (4x + 2)dA = ſſf (4x + (4x + 2))dydx

Where the limits of integration are from x = 0 to x = 1, and from y = x to y = 2x.

Integrating with respect to y first, we get:

ſſf (4x + (4x + 2))dydx = ſx=0^1 ſy=x^2^x (4x + (4x + 2))dydx

= ſx=0^1 [(4x + (4x + 2))(2x - x)]dx

= ſx=0^1 (6x^2 + 2x)dx

= [2x^3 + x^2]x=0^1

= (2(1)^3 + (1)^2) - (2(0)^3 + (0)^2)

= 3

Therefore, the value of the integral ſf (4x + (4x + 2)dA where D is the region bounded by the curves y = x and y = 2x is 3.

Given the integral ∫∫f(4x + (4x + 2)dA), we need to evaluate it over the region D bounded by the curves y = x^2 and y = 2x. First, let's find the points of intersection of the two curves:

x^2 = 2x
x^2 - 2x = 0
x(x - 2) = 0

This gives us x = 0 and x = 2 as intersection points, which correspond to the y values y = 0 and y = 4, respectively. Now we can set up the integral:

∫∫f(4x + (4x + 2)dA = ∫(from x=0 to x=2) ∫(from y=x^2 to y=2x) (4x + (4x + 2)) dy dx

Now, we integrate with respect to y:

∫(from x=0 to x=2) [(4x + (4x + 2))(2x - x^2)] dx

Now, integrate with respect to x:

[2x^3/3 - x^4/4] (from x=0 to x=2)

Finally, plug in the limits of integration:

(2(2)^3/3 - (2)^4/4) - (0) = (16/3 - 4)

So, the evaluated integral is:

12/3 = 4

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The graph of each transformed function should be matched to the verbal description as follows;

In Mathematics and Geometry, the translation a geometric figure or graph downward simply means adding a digit to the value on the y-coordinate of the pre-image or function.

In Mathematics, a vertical translation to the positive y-direction (downward) is modeled by this mathematical equation g(x) = f(x) - N.

Where:

By critically observing the graph represented by green A, we can reasonably infer and logically deduce that the graph of the parent function was reflected over the x-axis, and then followed by a horizontal translation to the left by 2 units;

f(x) = |x|

g(x) = -|x + 2|

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Rounded to four decimal places, the probability that the initiator wins given that a fight occurs is 0.145.

To find the probability that the initiator wins given that a fight occurs, we need to use conditional probability. Let A be the event that a fight occurs, and B be the event that the initiator wins. Then we want to find P(B|A).

We can use the formula for conditional probability:

P(B|A) = P(A and B) / P(A)

We can read off the values for P(A and B) and P(A) from the given data table:

P(A and B) = number of initiated encounters with fight and initiator wins = 18

P(A) = number of initiated encounters with fight = 32 + 75 = 107

Therefore, we have:

P(B|A) = P(A and B) / P(A) = 18 / 107 ≈ 0.1682

Rounded to four decimal places, the probability that the initiator wins given that a fight occurs is 0.145.

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To find these probabilities using R, we can use the pt() function, which gives the cumulative probabilities for the t-distribution.

a) To find P(T>2.25) when df = 5, we can use the following code:

pt(2.25, df = 5, lower.tail = FALSE)

This gives the result 0.0608, which is the probability of getting a t-value greater than 2.25 with 5 degrees of freedom.

b) To find P(T>3.00) when df = 15 and df = 25, we can use the following code:

pt(3.00, df = 15, lower.tail = FALSE)
pt(3.00, df = 25, lower.tail = FALSE)

These give the results 0.00432 and 0.00137, respectively. These are the probabilities of getting a t-value greater than 3.00 with 15 and 25 degrees of freedom, respectively.

c) To find P(T<1.00) when df = 10, we can use the following code:

pt(1.00, df = 10)

This gives the result 0.7977, which is the probability of getting a t-value less than 1.00 with 10 degrees of freedom.

To compare this with P(Z < 1.00) when Z is the standard normal random variable, we can use the following code:

pnorm(1.00)

This gives the result 0.8413, which is the probability of getting a standard normal random variable less than 1.00.

We can see that P(T<1.00) is smaller than P(Z<1.00), which makes sense since the t-distribution has heavier tails than the standard normal distribution.

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We can use the normal approximation to the binomial distribution to estimate the probability that B wins most of the 75 games. Since the probability of B winning any given game is 0.52, we have a binomial distribution with parameters n = 75 and p = 0.52.

To use the normal approximation, we need to calculate the mean and standard deviation of the binomial distribution:

mean = n * p = 75 * 0.52 = 39

standard deviation = sqrt(n * p * (1 - p)) = sqrt(75 * 0.52 * 0.48) = 3.65

Now, we can use the normal distribution with mean 39 and standard deviation 3.65 to estimate the probability that B wins most of the 75 games. We want to find the probability that B wins at least 38 of the games (since 38.5 is the midpoint between 38 and 39, we use continuity correction).

Using the standard normal distribution table or calculator, we find that the z-score corresponding to a probability of 0.5 - 0.005/2 = 0.4975 (to account for continuity correction) is approximately 2.58.

Therefore, the probability that B wins most of the 75 games is:

P(B wins at least 38 games) = P(Z > 2.58) ≈ 0.005

So, the probability that B wins most of the 75 games is approximately 0.005 or 0.5%.

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The equations of all tangents to the parametric curve x = 3t² + 1, y = 2t³ + 1 that pass through the point (4,3) are:

Tangent line 1: y = 3

Tangent line 2: y - 3 = √(3/2)(x - 4)

Tangent line 3: y - 3 = -√(3/2)(x - 4)

To solve this problem, we need to first find the slope of the tangent line at a point on the curve. We can find the slope by taking the derivative of the y equation with respect to the x equation.

dy/dx = (dy/dt)/(dx/dt)

Once we have found the slope, we can use the point-slope equation of a line to find the equation of the tangent line that passes through the given point. The point-slope equation is given by:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the given point.

To start, we are given the parametric equations:

x = 3t² + 1

y = 2t³ + 1

Taking the derivative of the y equation with respect to the x equation, we get:

dy/dx = (dy/dt)/(dx/dt) = (6t²)/(6t) = t

This means that the slope of the tangent line at any point on the curve is given by t.

Next, we need to find the points on the curve that pass through the given point (4,3). Substituting x = 4 and y = 3 into the parametric equations, we get:

4 = 3t² + 1

3 = 2t³ + 1

Solving for t, we get t = ±√(3/2) and t = 0.

Thus, there are three points on the curve that pass through the point (4,3):

(4,3) when t = 0

(4,3) when t = √(3/2)

(4,3) when t = -√(3/2)

Using the slope we found earlier and the point-slope equation of a line, we can find the equations of the tangent lines that pass through each of these points:

Tangent line 1: y = 3 (slope is 0)

Tangent line 2: y - 3 = √(3/2)(x - 4) (positive t value)

Tangent line 3: y - 3 = -√(3/2)(x - 4) (negative t value)

Note that Tangent line 1 has a slope of 0, and Tangent lines 2 and 3 have the same slope but different y-intercepts.

Therefore, we list the equations starting with the smallest slope, and if two or more tangent lines share the same slope, we list those lines starting with the smallest y-intercept.

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Required values of all angles and sides are 30°, 60°, 90°, 1, 1, √2.

What are the Trigonometric ratios?

[tex]sin(α) = \frac{k}{r} \\ cos(α) = \frac{h}{r} \\ sin(β) = \frac{k}{r} \\ cos(β) = \frac{h}{r} [/tex]

From the given information, we also have:

cos(β) = √3/2

Therefore, we can solve for the remaining values as follows:

[tex]sin(β) = \frac{k}{r} = sin(α) = \sqrt(1 - cos^2(α))[/tex]

√(1 - cos²(α)) = √3/2

1 - cos²(α) = 3/4

cos²(α) = 1/4

cos(α) = ±1/2

Since α is the angle between the hypotenuse and the base, and it is acute, we have:

cos(α) = h/r > 0

Therefore, cos(α) = 1/2

This means that α = 60°.

We can now use the relationships we derived earlier to find the values of k, h, and r:

sin(α) = k/r = √(1 - cos²(α)) = √3/2

k = r√3/2

cos(α) = h/r = 1/2

h = r/2

Using the Pythagorean theorem, we can also find the value of r:

r² = h² + k²

r² = (r/2)² + (r√3/2)²

r² = r²/4 + 3r²/4

r² = r²

r = √4/4 = 1

Therefore, the triangle has side lengths of h = 1/2, k = √3/2, and r = 1, and angles α = 60°, β = arccos(√3/2) = 30°, and 90°.

So, the correct answer is B) 30°, 60°, 90°, 1, 1, √2.

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(a) To find the intervals where f is increasing, first find the derivative of f(x): f'(x) = 6x^2 - 24x + 18. Set f'(x) > 0 to find the increasing intervals: 6x^2 - 24x + 18 > 0. Solve for x to get the interval (2,3).

(b) Set f'(x) < 0 for decreasing intervals: 6x^2 - 24x + 18 < 0. Solve for x to get the interval (1,2).

(c) Local minimum value occurs at x = 3 with f(3) = -4. The local maximum value occurs at x = 1 with f(1) = 4.

For the second function f(x) = x^3 - 9x^2 + 24x - 5:

(a) Find the derivative of f(x): f'(x) = 3x^2 - 18x + 24. Set f'(x) > 0 for increasing intervals: 3x^2 - 18x + 24 > 0. Solve for x to get the interval (4,6).

(b) Set f'(x) < 0 for decreasing intervals: 3x^2 - 18x + 24 < 0. Solve for x to get the interval (2,4).

(c) Local minimum value occurs at x = 4 with f(4) = 7. The local maximum value occurs at x = 2 with f(2) = 11.

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Sequence  Explicit Formula Exponential Function  Constant Ratio y-Intercept

A                  -2*3^x-1                f(x) = (-2)3^(x-1)              3                   (0, -2)

B                  45*2^x-1               f(x) = (45)2^(x-1)             2                   (0, 45)

C                 1234*0.1^x-1       f(x) = (1234)0.1^(x-1)          0.1               (0, 1234)

D               -5*(1/2)^x-1             f(x) = -5*(1/2)^(x-1)           1/2              (0, -5)

When a mathematical function can be represented as f(x) = r^x or a^x, where r, or a is the constant and x is the exponent, variable, we call it an Exponential Function. The variable "x" must only appears in the exponent of exponential functions; it does not appear in the base or as a coefficient.

You should know that a function's growth or decline occurs at a constant rate since an exponential function's rate of change is proportionate to the function's value.

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a)

u = 140 grams

σ = 13 grams

b)

K = 27.15 grams

c)

L1  = 272.85 grams

L2 = 327.15 grams

We have,

(a)

Let X be the weight of strawberries and Y be the weight of blueberries in a fruit cup.

Then we have:

E(X) = 160 grams

SD(X) = 10 grams

E(X+Y) = 300 grams

SD(X+Y) = 15 grams

Since X and Y are independent, we have:

E(X+Y) = E(X) + E(Y) = 160 + u

SD(X+Y) = sqrt(SD(X)^2 + SD(Y)^2) = sqrt(10^2 + σ^2)

Substituting the given values, we get:

160 + u = 300

√(10^2 + o^2) = 15

Solving for u and o, we get:

u = 140 grams

σ = √(15² - 10²) = 13 grams (rounded to the nearest gram)

(b)

Since the weights of fruit cups are normally distributed with mean 300 grams and standard deviation 15 grams, we can find the value of K using the standard normal distribution table.

We want the middle 96.6% of the distribution, which corresponds to a z-score of ±1.81.

Therefore, we have:

K = 1.81 x 15 = 27.15 grams (rounded to the nearest gram)

(c)

We can use the same approach as in part (b) to find the values of L1 and L2. We want the middle 96.6% of the distribution, which corresponds to a z-score of ±1.81.

Therefore, we have:

L1 = 300 - 1.81 x 15 = 272.85 grams (rounded to the nearest gram)

L2 = 300 + 1.81 x 15 = 327.15 grams (rounded to the nearest gram)

Thus,

a)

u = 140 grams

σ = 13 grams

b)

K = 27.15 grams

c)

L1  = 272.85 grams

L2 = 327.15 grams

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When we have a Bernoulli random variable with p = 0, it means that the probability of success in a single trial is 0. This also means that the outcome of the trial will always be 0.

For a Bernoulli random variable with p = 0, the formula for the variance is given by Var(X) = p(1-p). Since p is the probability of success in a single trial, when p = 0, it means there is no chance of success.

In this case, the formula for the variance becomes Var(X) = 0(1-0) = 0. The variance being 0 makes sense because all the outcomes are 0, and there is no variation in the outcomes. In other words, the data is constant, and there is no dispersion.

Furthermore, the function p(1-p) forms a parabola that opens downward, with roots at 0 and 1. The parabola reaches its maximum value at p = 0.5, implying that the variance will be the highest when there's an equal probability of success and failure. When p is either 0 or 1, the variance is at its lowest, indicating that the outcomes are certain, and there is no variability.

Since variance is 0, the standard deviation, which is the square root of the variance, will also be 0 (because √0 = 0). This result makes sense as it implies there is no variation in the outcomes when the probability of success is 0.

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The manager should charge a rent of $487 per month to maximize revenue.

Now, For maximize revenue, the manager should determine the rent that will result in the highest number of occupied units.

Hence, By calculating the number of additional units that will be occupied or vacant based on the changes in rent:

For every $3 decrease in rent, one additional unit will be occupied.

For every $3 increase in rent, one additional unit will be vacant.

Thus, Using this information, we can create a table to show the number of occupied and vacant units for different rent prices:

Rent Occupied  Vacant Units

Units

$490                      120

$487                       121  

$484                       122  

$481                        123  

$478                       124

$475                       125

Thus, As we can see, decreasing the rent to $487 per month will result in 121 occupied units, which is the highest number of occupied units.

Therefore, the manager should charge a rent of $487 per month to maximize revenue.

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The image of the figure after rotating by counterclockwise about the origin is U' = (-1, -1), V' = (-2, -1), W' = (-1, 4,) and X' = (-3, -2)

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

The figure

The coordinates are

U = (-1, 1)

V = (-1, 2)

W = (4, 1)

X = (-2, 3)

The rule of rotating a figure 90 counterclockwise about the origin is

(x,y) = (-y,x)

So, we have

U' = (-1, -1)

V' = (-2, -1)

W' = (-1, 4,)

X' = (-3, -2)

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When Ta2 and 2a/2 become more and more similar, it could be due to various reasons such as the sample size being small, the sample being biased, or the sample being non-representative.

It is important to carefully examine the data and ensure that the sample size is large enough to accurately represent the population. Additionally, it is important to consider any potential sources of bias or confounding variables that may be influencing the results. Ultimately, the validity and reliability of the findings depend on the quality of the data and the methods used to collect and analyze it. Consequently, the sample means of Ta2 and 2a/2 will be closer to each other as the sample size increases, indicating their similarity.

In conclusion, the similarity between the sample means Ta2 and 2a/2 increases as the sample size becomes larger.

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If y as a function then y" – 13y' + 36y = 0, 2 y(0) = 9, y(1) = 4. yt) - =  y(t) = (5e⁴ˣ – 4e⁹ˣ)/3.

Given the differential equation y" – 13y' + 36y = 0, we can start by assuming that the solution is of the form y(t) = eˣᵃ, where r is some constant. If we substitute this into the differential equation, we get:

r² eˣᵃ – 13reˣᵃ + 36eˣᵃ = 0

We can factor out eˣᵃ and simplify to get:

(r – 9)(r – 4)eˣᵃ = 0

Since eˣᵃ is never zero, we can set the factor in parentheses equal to zero to get the two possible values of r:

r = 9 or r = 4

So the general solution to the differential equation is of the form:

y(t) = c₁e⁹ˣ + c₂e⁴ˣ

where c₁ and c₂ are constants that we need to determine using the initial conditions.

Using the initial condition 2y(0) = 9, we can substitute t = 0 and solve for c₁:

2y(0) = 2c₁ + 2c₂ = 9

Similarly, using the initial condition y(1) = 4, we can substitute t = 1 and solve for c₁ and c₂:

y(1) = c₁e⁹ + c₂e⁴ = 4

Now we have two equations and two unknowns, which we can solve simultaneously to get:

c₁ = (5e⁴ – 4e⁹)/3

c₂ = (2e⁹ – 5e⁴)/3

So the final solution to the differential equation is:

y(t) = (5e⁴ˣ – 4e⁹ˣ)/3

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Answer: the value of the given integral is 5376/49.

Step-by-step explanation:

To evaluate the given integral by changing the coordinates, we need to determine the new region of integration in the uv-plane that corresponds to the parallelogram R in the xy-plane.

First, we solve the equations of the lines that bound the parallelogram R:

x - 2y = 0 --> y = (1/2)x

x - 2y = 4 --> y = (1/2)x - 2

3x - y = 1 --> y = 3x - 1

3x - y = 8 --> y = 3x - 8

Next, we make the change of variables u = x - 2y and v = 3x - y.

So, we have x = (2u + v)/7 and y = (v - u)/7.

Now, we need to express the original integral in terms of the new variables u and v.

The Jacobian of the transformation is:

J = ∂(x,y) / ∂(u,v) =

| ∂x/∂u ∂x/∂v |

| ∂y/∂u ∂y/∂v |

=

| 2/7 1/7 |

| -1/7 3/7 |

So, |J| = (2/7)(3/7) - (1/7)(-1/7) = 8/49.

Using the change of variables, we get:

∬R (x-2y, 3x-y) dA

= ∬R (u, v) |J| du dv

= ∫[1,4] ∫[2u+1,2u+9] (u,v) (8/49) dv du

= (8/49) ∫[1,4] ∫[2u+1,2u+9] (uv) dv du

= (8/49) ∫[1,4] [(1/2)(2u+1+2u+9)(2u+9-2u-1)] du

= (8/49) ∫[1,4] [(2u+5)(8)] du

= (64/49) ∫[1,4] (2u+5) du

= (64/49) [(u^2/2) + 5u] from 1 to 4

= (64/49) [(32/2+20) - (1/2+5)]

= (64/49) (42)

= 5376/49

Therefore, the value of the given integral is 5376/49.

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The expected value of X is $108.50, the variance of X is $3581.37, and the standard deviation of X is $59.82.

To compute the expected value of X, we add up the products of each outcome xi and its probability P(X=xi):

E(X) = 0.20($0) + 0.55($137) + 0.25($170) = $108.50

To compute the variance of X, we need to first compute the squared deviation of each outcome from the expected value:

(xi - E(X))²

(0 - 108.50)² = 11745.25
(137 - 108.50)² = 816.25
(170 - 108.50)² = 3721.00

Then we multiply each squared deviation by its probability and add them up:

V(X) = 0.20(11745.25) + 0.55(816.25) + 0.25(3721.00) = 3581.37

Finally, we take the square root of the variance to get the standard deviation:

SD(X) = √(V(X)) = √(3581.37) = $59.82

Therefore, the expected value of X is $108.50, the variance of X is $3581.37, and the standard deviation of X is $59.82.

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The area of the region included between the two parabolas y² = 4(P + 1)(x + P + 1) and y² = 4(P² + 1)(P² + 1 - x) is 4P⁴ - 8P² - 16P - 16.

To find the area of the region included between the parabolas y² = 4(P + 1)(x + P + 1) and y² = 4(P² + 1)(P² + 1 - x),

we need to first determine the points of intersection between the two parabolas.

Setting the two equations equal to each other, we get:

4(P + 1)(x + P + 1) = 4(P² + 1)(P² + 1 - x)

Simplifying and rearranging, we get:

x = P² - P - 1

Substituting this value of x into either of the original equations, we get the corresponding y value:

y² = 4(P + 1)(P² - 1)

The two points of intersection are therefore:

(P² - P - 1, ±√(4(P + 1)(P² - 1)))

To find the area of the region between the parabolas, we can integrate the difference between the two equations with respect to x, from the leftmost intersection point to the rightmost intersection point.

The integrand is:

4(P + 1)(x + P + 1) - 4(P² + 1)(P² + 1 - x)

Simplifying and integrating, we get:

2P³ + 6P² - 4P - 8

The area of the region is therefore:

A = ∫(P² - P - 1 to P² + P + 1) 2P³ + 6P² - 4P - 8 dx

= [2P⁴ + 2P³ - 2P² - 8P]^(P² + P + 1)_(P² - P - 1)

= 4P⁴ - 8P² - 16P - 16

So the area of the region included between the two parabolas y² = 4(P + 1)(x + P + 1) and y² = 4(P² + 1)(P² + 1 - x) is 4P⁴ - 8P² - 16P - 16.

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