when+the+temperature+of+a+copper+penny+is+increased+by+100+c°,+its+diameter+increases+by+0.17%.+the+area+of+one+of+its+faces+increases+by:+

Answer:

v=l × b×h

= 8×6×5

=240 units³

The solution to the system of equations is {x = -1, y = -7/2}.

1.3.1 Solve for x: (3x - 1)(x + 2) = 7x + 5

First, let's expand the left-hand side of the equation:

3x^2 + 6x - x - 2 = 7x + 5

Simplify to:

3x^2 + 5x - 2 = 7x + 5

Subtract 7x + 5 from both sides to set the equation to zero:

3x^2 - 2x - 7 = 0

This is a quadratic equation in the form [tex]ax^2 + bx + c = 0[/tex]. To solve it, we can use the quadratic formula, x = [-b ± [tex]\sqrt(b^2 - 4ac)] / 2a:[/tex]

[tex]x = [2 \sqrt((-2)^2 - 43(-7))] / 2*3\\x = [2 \sqrt(4 + 84)] / 6\\x = [2 \sqrt(88)] / 6\\\\x = [2 2\sqrt(22)] / 6\\x = 1/3 \sqrt(22)/3[/tex]

So the solution set for this equation is {x = 1/3 + sqrt(22)/3, x = 1/3 - sqrt(22)/3}.

1.3.2 Solve for x: 2x - 5 ≥ -(x + 1)

First, simplify the inequality:

2x - 5 ≥ -x - 1

Add x and 5 to both sides to isolate x:

3x ≥ 4

Divide by 3:

x ≥ 4/3

So the solution set for this inequality is {x | x ≥ 4/3}.

1.4 Solve for x and y simultaneously in: 6 + 2y - x = 0 and 3x - 2y - 4 = 0

Rearrange the first equation to x = 6 + 2y and substitute into the second equation:

3(6 + 2y) - 2y - 4 = 0

18 + 6y - 2y - 4 = 0

4y + 14 = 0

4y = -14

y = -14/4

y = -7/2

Substitute y = -7/2 into the first equation:

6 + 2(-7/2) - x = 0

6 - 7 - x = 0

-x = 1

x = -1

So the solution to the system of equations is {x = -1, y = -7/2}.

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Two possible expressions for the length and the width of the rectangle are:

Remember that for a rectangle of length L and width W, the area is:

A = L*W

Here the area is given by the quadratic equation:

A = x² + 2x - 7x - 14

We can factorize this equation to get:

A = (x + 2)(x - 7)

Then we can define:

length = x + 2

width = x- 7

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To represent b^11 as a product of terms of the form b^2, we can write it as (b^2)^5. This means that b^11 can be expressed as the product of five terms of the form b^2.

b^11 = (b^2) * (b^2) * (b^2) * (b^2) * (b^2)

In this representation, each term is b^2, and we have a total of five terms. By multiplying these terms together, we get b^11.

The exponent rule states that when we raise a power to another power, we multiply the exponents. In this case, (b^2)^5 means we multiply the exponent 2 by 5, resulting in b^10. Therefore, the product of five terms of the form b^2 gives us b^11.

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The area of a regular hexagon can be calculated using the formula A = (3√3/2) * s^2, where A is the area and s is the length of each side.

In this case, the hexagon has a radius of 5 inches, so the length of each side can be found by multiplying the radius by 2 and dividing it by the square root of 3. Substituting the side length into the formula gives the area of the regular hexagon.

The formula for calculating the area of a regular hexagon is derived from its relationship to an equilateral triangle. The regular hexagon can be divided into six equilateral triangles, where the side length of the hexagon is equal to the base of each equilateral triangle.

The formula A = (3√3/2) * s^2 is a result of the area formula for an equilateral triangle, where s is the length of each side. In this case, the radius of the hexagon is given as 5 inches. To find the length of each side, we multiply the radius by 2 and divide it by the square root of 3. Substituting this value into the formula gives us the area of the regular hexagon.

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The correct answer is:  C) 0.  If adding an additional input does not result in any increase in output, it suggests that the production function has reached a point of diminishing returns or a maximum level of productivity. At this point, the slope of the production function is zero.

The slope of a production function represents the rate at which output changes with respect to changes in input. A positive slope indicates increasing output with additional input, while a negative slope implies decreasing output. However, a zero slope indicates that there is no change in output despite changes in input.

In this scenario, since adding an additional input does not generate any additional output, the production function has plateaued, and the slope is zero. This means that the production function has reached its maximum level of efficiency or capacity.

Therefore, the correct answer is:

C) 0

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Two fair dice are tossed, and the up face on each die is recorded.

The probability of observing each of the given events: are

P(A) = 1/3

P(B) = 5/18

P(C) = 1/2

A: The difference of the numbers is 2 or less.

The favorable outcomes for event A are when the difference between the numbers on the dice is 2 or less. We can have the following outcomes:

(1,1), (1,2), (2,1), (2,2), (1,3), (3,1), (2,3), (3,2), (3,3), (4,4), (5,5), (6,6)

So, there are 12 favorable outcomes for event A.

The total number of possible outcomes when two dice are tossed is 6 * 6 = 36.

Therefore, the probability of event A, P(A), is 12/36 = 1/3.

B: A 6 appears on exactly one of the dice.

The favorable outcomes for event B are when a 6 appears on exactly one of the dice. We can have the following outcomes:

(6,1), (6,2), (6,3), (6,4), (6,5), (1,6), (2,6), (3,6), (4,6), (5,6)

So, there are 10 favorable outcomes for event B.

Again, the total number of possible outcomes is 6 * 6 = 36.

P(B), is 10/36 = 5/18.

Therefore, the probability of event B,

C: The sum of the numbers is even.

The favorable outcomes for event C are when the sum of the numbers on the dice is even. We can have the following outcomes:

(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2), (6,4), (6,6)

So, there are 18 favorable outcomes for event C.

Once again, the total number of possible outcomes is 6 * 6 = 36.

Therefore, the probability of event C, P(C), is 18/36 = 1/2.

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To calculate the interest earned by Maria in 3 years, we can use the simple interest formula:

Interest = Principal * Rate * Time

Given:

Principal (P) = $2,300

Rate (R) = 5% = 0.05 (as a decimal)

Time (T) = 3 years

Substituting these values into the formula, we have:

Interest = $2,300 * 0.05 * 3

Calculating the result:

Interest = $2,300 * 0.05 * 3

= $345

Therefore, Maria will earn $345 in interest over 3 years.

∂w/∂r = (y * yz * zx) * (∂x/∂r) + (x * yz * zx) * (∂y/∂r) + (x * y * zx) * (∂z/∂r)

∂w/∂θ = (y * yz * zx) * (∂x/∂θ) + (x * yz * zx) * (∂y/∂θ) + (x * y * zx) * (∂z/∂θ)

To find the partial derivative ∂w/∂r, we use the chain rule. We differentiate each term in the expression for w with respect to r, while considering the chain rule for each variable. Since x = r * cos(θ), y = r * sin(θ), and z = r, we find the partial derivatives (∂x/∂r), (∂y/∂r), (∂z/∂r), (∂x/∂θ), (∂y/∂θ), and (∂z/∂θ).

For ∂w/∂r, we differentiate each term with respect to r, resulting in (y * yz * zx) * cos(θ) + (x * yz * zx) * sin(θ) + (x * y * zx). Similarly, for ∂w/∂θ, we differentiate each term with respect to θ, resulting in (-y * yz * zx) * r * sin(θ) + (x * yz * zx) * r * cos(θ).

Given that r = 4 and θ = 2, we substitute these values into the respective expressions to obtain the numerical values for ∂w/∂r and ∂w/∂θ.

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In a situation where the dependent variable can assume only one of two possible discrete values, logistic regression should be applied. Therefore, the correct choice is option c.

When the dependent variable is binary or dichotomous, meaning it can take only two possible discrete values (such as "yes" or "no," "success" or "failure," etc.), logistic regression is the appropriate statistical technique to analyze the data. Logistic regression is specifically designed for modeling binary outcomes.

Multiple regression, option a, is not necessary in this case because the dependent variable is not continuous, and multiple regression is typically used when the dependent variable is continuous.

Option b, stating that all the independent variables must have values of either zero or one, is not universally true for all situations with a binary dependent variable. The values of independent variables in logistic regression can take various forms, including continuous, categorical, or binary.

Option d, claiming that there can only be two independent variables, is incorrect. The number of independent variables in logistic regression is not restricted to two; it can involve multiple independent variables, similar to multiple regression.

Therefore, the correct choice is option c: logistic regression should be applied when the dependent variable can assume only one of two possible discrete values.

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The independent variable is h, the number of hours Vincent spends reading every day, and the dependent variable is d, the number of days it will take Vincent to finish reading the book.

The independent variable is the variable that can be changed by the experimenter. In this case, Vincent can change the number of hours he spends reading every day. The dependent variable is the variable that is affected by the independent variable. In this case, the number of days it will take Vincent to finish reading the book depends on the number of hours he spends reading every day. For example, if Vincent spends 2 hours reading every day, it will take him 15 days to finish the book. If he spends 3 hours reading every day, it will take him 10 days to finish the book. The number of hours he spends reading every day (the independent variable) determines the number of days it will take him to finish the book (the dependent variable).

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The equation for the Unknown logo side length maximum area is : A = (21/2) × W.

To determine the unknown side length, L, of the rectangular logo, we can set up an equation using the given information. Let's assume the width of the logo is W.

The area of a rectangle is given by the formula: A = length × width.

In this case, the area is said to be exactly the maximum area allowed by the building owner. So, we need to maximize the area, given the constraint that the height of the logo is 21/2 feet.

The equation for the area is: A = L × W.

From the given information, we know that the height (L) of the logo is 21/2 feet.

Therefore, the equation for the Unknown logo side length maximum area is : A = (21/2) × W.

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Answer:

Step-by-step explanation: A) Relative frequency is the number of times an event happens over a total number of events :

so, 23/50 = 0.23

B) As there's a 6 side on a die so there are 2 even numbers and 3 odd numbers, so the theoretical probability of landing on an even number would be 3/6 = 0.50

c)Since the die has an equal distribution of chance landing of an even or a odd number (3/6 for both ), hence the die is not biased.

To find the flux of the vector field F = (0, 0, 3) across the slanted face of the tetrahedron, we need to calculate the surface integral over the region R in the xy-plane.

The equation of the slanted face of the tetrahedron is given by z = 5 - x - y. To determine the limits of integration for the double integral, we need to find the projection of the region R onto the xy-plane.

By setting z = 0 in the equation of the slanted face, we can solve for the corresponding values of x and y:

0 = 5 - x - y

x + y = 5

This equation represents a straight line in the xy-plane passing through the points (5, 0) and (0, 5). This line determines the bounds for the double integral.

The flux integral can be set up as follows:

Flux = ∬_R F · n dA

Here, F = (0, 0, 3) is the vector field, and n is the outward unit normal vector to the surface. Since the normal vectors point upward, we can take n = (0, 0, 1).

The double integral over the region R in the xy-plane becomes:

Flux = ∬_R (F · n) dA

    = ∬_R (0, 0, 3) · (0, 0, 1) dA

    = ∬_R 3 dA

Since the integrand is a constant, we can evaluate the double integral by finding the area of region R in the xy-plane and multiplying it by the constant:

Flux = 3 * Area(R)

To determine the area of region R, we can calculate the area of the triangle formed by the line x + y = 5. The vertices of this triangle are (0, 5), (5, 0), and the origin (0, 0).

Using the formula for the area of a triangle, we have:

Area(R) = (1/2) * base * height

        = (1/2) * 5 * 5

        = 12.5

Therefore, the flux of the vector field across the slanted face of the tetrahedron is given by:

Flux = 3 * Area(R)

    = 3 * 12.5

    = 37.5

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If the function g(x) is exponential and it increases by a factor of 3 over every unit interval, we can express it in the form g(x) = a * 3^x, where 'a' is a constant.

Given that g(x) has the value 9 when x = 6, we can substitute these values into the equation:

9 = a * 3^6

To solve for 'a', we divide both sides of the equation by 3^6:

a = 9 / 3^6 = 9 / 729 = 1/81

Therefore, the function g(x) is given by g(x) = (1/81) * 3^x, where it increases by a factor of 3 over every unit interval and has the value 9 when x = 6.

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The given limit represents the derivative f′(a). The function f(x) can be determined by finding the antiderivative of the derivative function f′(x), and the value of a can be calculated by evaluating the given limit expression.

To find the function f(x), we need to find the antiderivative of f′(x). In this case, f′(x) is represented by cos((π/2)+h)/h. Integrating this function will give us f(x) up to an arbitrary constant. However, since the question asks us to find f(x) in terms of x, we can write f(x) as the definite integral from a constant c to x of f′(t) dt, where f′(t) is the given derivative function.

To calculate the value of a, we evaluate the given limit expression as h approaches 0. Plugging in h = 0 into the expression cos((π/2)+h)/h will result in an indeterminate form of 0/0. This suggests the application of L'Hôpital's rule, which states that for indeterminate forms, taking the derivative of the numerator and denominator and then evaluating the limit can often yield a determinate form. By applying L'Hôpital's rule, we differentiate the numerator and denominator separately and re-evaluate the limit.

In conclusion, finding f(x) requires integrating the given derivative function, and calculating the value of a involves using L'Hôpital's rule to evaluate the given limit expression.

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The accounts receivable turnover for Danielle Manufacturing Company is 8.3 times. This indicates that on average, the company collects its accounts receivable 8.3 times throughout the year.

To calculate the accounts receivable turnover, we divide the net sales by the average accounts receivable. The average accounts receivable can be calculated by adding the beginning and ending accounts receivable and dividing the sum by 2.

In this case, the average accounts receivable is ($60,000 + $90,000) / 2 = $75,000.

Now, we divide the net sales of $750,000 by the average accounts receivable of $75,000 to get the accounts receivable turnover:

Accounts Receivable Turnover = Net Sales / Average Accounts Receivable

                                = $750,000 / $75,000

                                = 10 times.

Therefore, the correct answer is 10 times, not 8.3 times as initially stated.

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Answer: Mateo's required minimum distribution is $14,736.60.

Step-by-step explanation: RMD = IRA balance / life expectancy factor

Plugging in the numbers given:

RMD = $390838.04 / 26.5

RMD = $14,736.60

The bond interest expense reported on the income statement for the end of the first year should be approximately $421,480.

For calculating the bond interest expense for the end of the first year using the effective interest method of amortization:

1: Determine the effective interest rate:

The effective annual rate is given as 8%. Since the bonds were issued at a discount (purchase price of $5,268,500 is less than the face value of $6,100,000), the effective interest rate will be higher than the coupon rate of 6%.

The effective interest rate is used to calculate interest expense over the life of the bond.

2: Calculate the annual interest expense:

The annual interest expense is calculated by multiplying the carrying value of the bond at the beginning of the period by the effective interest rate.

Carrying value = Face value of the bond - Accumulated amortization

For the first year, the carrying value at the beginning of the period is the same as the purchase price since no amortization has been recorded yet.

Annual interest expense = Carrying value at the beginning of the period * Effective interest rate

3: Calculate the amortization:

Amortization is the difference between the annual interest expense and the coupon payment. The difference is added to the bond carrying value, reducing the discount.

Amortization = Annual interest expense - Coupon payment

Finally, we can calculate the bond interest expense for the end of the first year:

1. Calculate the annual interest expense:

Annual interest expense = Carrying value at the beginning of the period * Effective interest rate

2. Calculate the amortization:

Amortization = Annual interest expense - Coupon payment

3. Calculate the bond interest expense:

Bond interest expense = Coupon payment + Amortization

Let's calculate these values using the information provided:

Face value of the bond = $6,100,000

Purchase price of the bond = $5,268,500

Coupon rate = 6%

Effective annual rate = 8%

Number of years = 10

1: Determine the effective interest rate:

Effective interest rate = 8%

2: Calculate the annual interest expense:

Carrying value at the beginning of the period = Purchase price = $5,268,500

Annual interest expense = Carrying value at the beginning of the period * Effective interest rate

Annual interest expense = $5,268,500 * 8% = $421,480

3: Calculate the amortization:

Coupon payment = Face value of the bond * Coupon rate

Coupon payment = $6,100,000 * 6% = $366,000

Amortization = Annual interest expense - Coupon payment

Amortization = $421,480 - $366,000 = $55,480

4: Calculate the bond interest expense:

Bond interest expense = Coupon payment + Amortization

Bond interest expense = $366,000 + $55,480 = $421,480

Therefore, the bond interest expense reported on the income statement for the end of the first year should be approximately $421,480.

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P(X > 150) = P(Z > (150 - 400/3) / (20/3))

where Z is a standard normal random variable.

What is Probability?

Probability is a branch of mathematics concerned with numerical descriptions of how likely an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates the impossibility of the event and 1 indicates a certainty

a) To calculate the probability that tails shows up for the first time at the 10th flip, we need to consider the sequence of flips leading up to the 10th flip.

The probability of getting tails on a single flip is 1/3, and the probability of getting heads is 2/3. Since the coin flips are independent events, the probability of getting tails on the first nine flips and then heads on the 10th flip is:

(1/3)^9 * (2/3) = 2^-9 * 3^-9

This is because the probability of getting tails on each of the nine flips is (1/3)^9, and the probability of getting heads on the 10th flip is 2/3.

Therefore, the probability that tails shows up for the first time at the 10th flip is approximately:

2^-9 * 3^-9 = 1/19683 ≈ 0.000051

b) To calculate the probability of more than 150 heads showing up using a suitable approximation, we can make use of the normal approximation to the binomial distribution.

In this case, we have 200 coin flips with a probability of heads occurring in each flip as 2/3. The expected number of heads is given by the product of the number of flips (200) and the probability of heads (2/3):

Expected number of heads = 200 * (2/3) = 400/3

The standard deviation of a binomial distribution is given by the square root of the product of the number of flips, the probability of success, and the probability of failure:

Standard deviation = sqrt(200 * (2/3) * (1/3)) = sqrt(400/9) = 20/3

To find the probability of more than 150 heads, we can approximate it as the probability of the number of heads being greater than 150 in a normal distribution with a mean of 400/3 and a standard deviation of 20/3.

Using a standard normal distribution table or a calculator, we can calculate the probability:

P(X > 150) = P(Z > (150 - 400/3) / (20/3))

where Z is a standard normal random variable.

By substituting the values and evaluating the expression, we can find the probability more than 150 heads shows up.

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Variance standard deviation coefficient of variation mean:

(a) The measures of variation that indicate spread about the mean are variance and standard deviation. Variance is the average squared deviation from the mean and provides an estimate of the degree of spread or dispersion of the data. Standard deviation is the square root of variance and is a commonly used measure of the spread of data. Coefficient of variation is also a measure of variation, which expresses the standard deviation as a percentage of the mean.

(b) The graphic display that shows the median and data spread about the median is the box-and-whisker plot. The box-and-whisker plot displays the five-number summary, which includes the minimum value, the first quartile, the median, the third quartile, and the maximum value. The box represents the middle 50% of the data and the whiskers show the range of the data outside the box. The median is represented by a line inside the box. The box-and-whisker plot is a useful tool for comparing distributions and identifying outliers.

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The additional amount she needs to buy the new car is $18,027

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

The cost of the new car = $25,025.

The worth of the car now =  $6998.

Using the above as a guide, we have the following:

The amount needed is the difference between the above costs

So, we have

Difference = 25,025 - 6998

Evaluate

Difference = 18027

Hence, she needs $18,027 to buy the new car

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Answer:

i hope this halp you

Step-by-step explanation:

To find the area of the surface that lies above the rectangle [0, 7] × [1, 8] and below the plane z = 6 + 2x + 5y, we can use double integration.

The surface is defined by the equation z = 6 + 2x + 5y. To find the area of this surface, we need to integrate over the rectangular region [0, 7] × [1, 8]. We can set up a double integral in terms of x and y to calculate the surface area.

The integral for the surface area is given by A = ∬R √(1 + (∂z/∂x)^2 + (∂z/∂y)^2) dA, where R represents the rectangular region [0, 7] × [1, 8], ∂z/∂x and ∂z/∂y represent the partial derivatives of z with respect to x and y, respectively, and dA represents the differential area element.

To evaluate the integral, we calculate the partial derivatives (∂z/∂x and ∂z/∂y), substitute them into the integrand, and integrate over the rectangular region R. This will yield the area of the surface that lies above the given rectangle.

Performing the necessary calculations and evaluating the double integral will give us the area of the surface.

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Yes, v is an eigenvector of the matrix A.

No, v is not an eigenvector of the matrix A.

To determine if v is an eigenvector of matrix A, we need to check if the following equation holds: Av = λv, where λ is a scalar called the eigenvalue.

For the first question, we have A = [[-3, -4, 6], [-2, -2, 6], [-4, -7, 20]], and v = [-1, 1, 0]. Multiplying Av, we get Av = [-2, 2, 0], and multiplying λv, we get λv = [-λ, λ, 0]. To find the eigenvalue λ, we solve the equation Av = λv, which leads to λ = 2. Since Av = λv, we can conclude that v is an eigenvector of A.

For the second question, we have A = [[1, 2], [3, 4]], and v = [-1, 1]. Multiplying Av, we get Av = [-1, 1], and multiplying λv, we get λv = [-λ, λ]. To find the eigenvalue λ, we solve the equation Av = λv, which leads to λ = 3 or λ = -1. Since neither λ = 3 nor λ = -1 makes Av = λv true, we can conclude that v is not an eigenvector of A.

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In the completed column of Joan's preference schedule, Carson Wentz received 25 votes for 1st place, Nick Foles received 50 votes for 2nd place, Nelson Agholor received 100 votes for 3rd place, and Jason Kelce received 0 votes for 4th place. The total number of votes in this column is 175.

Joan surveyed her friends online to determine their preferences for meeting current Philadelphia Eagles football players. She asked them to rank the players from first to fourth. Joan is using the Borda count method to create her preference schedule. The preference schedule is as follows:

1st place:

Carson Wentz received 25 votes.

2nd place:

Nick Foles received 50 votes.

3rd place:

Nelson Agholor received 100 votes.

4th place:

Jason Kelce received 0 votes.

In total, there were 175 votes cast by Joan's friends.

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The problem provides a decay law for the radioactive element polonium, where the amount of the element left after time t is given by Q(t) = Q0 · 2-(t/140), where Q0 is the initial amount.

The question asks us to find the initial amount of polonium present given that 10 mg of the element is left after 700 days. To solve this problem, we can substitute the given values into the decay law and solve for Q0. We can write the equation as 10 = Q0 · 2^(-700/140), and then simplify to 10 = Q0 · 2^(-5), or Q0 = 10 · 2^5 = 320 mg.

In summary, the problem provides a decay law for polonium and asks us to find the initial amount of the element given the amount left after a certain amount of time. By substituting the given values into the decay law and solving for Q0, we find that the initial amount of polonium present was 320 mg.

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Answer: x= 11 43/45 or decimal form x=-11.9

Step-by-step explanation:

12(8-4×)+3×=634

You start by

12x(4x)+3x=634

48x+3x=634

51x=634

634 divided 51=12.431372549

X=12.431372549

The confidence interval for the population mean is 72.05036, 90.38714

Confidence interval = sample mean ± (critical value) × (standard deviation / √(sample size))

Test scores of 32 students at a 99% confidence level.

Mean = Summing of all the test scores /sample size

80 + 74 + 61 + 93 + 96 + 70 + 80 + 64 + 51 + 98 + 93 + 87 + 72 + 77 + 84 + 96 + 100 + 67 + 71 + 79 + 99 + 85 + 66 + 70 + 57 + 75 + 86 + 92 + 94 + 70 + 81 + 89 = 2599

Sample mean = 2599 / 32 = 81.21875

For standard deviation

The sum of squared deviations from the sample mean:

(80 - 81.21875)² + (74 - 81.21875)² + ... + (89 - 81.21875)² =

divide the sum by the sample size -1 and take the square root

√(12774.5625 / (32 - 1)) = √(400.4545455) = 20.011

Standard deviation = 20.011

The critical value for a 99% confidence level is approximately 2.617.

Putting the values into the formula

Confidence interval = 81.21875 ± (2.617) × (20.011 / √(32))

Calculating the square root of the sample size

√(32) = 5.656854249

Confidence interval = 81.21875 ± (2.617) × (20.011 / 5.656854249)  Confidence interval = 81.21875 ± 9.16839

The lower bound of the confidence interval is approximately 72.05036, and the upper bound is approximately 90.38714.

Therefore, the 99% confidence interval for the population mean is approximately (72.05036, 90.38714).

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b)The marginal PDF fx(x) is given by: fx(x) = ([tex]2\sqrt{(r^2 - x^2))/(\pi r^2}[/tex]), for -r < x <r

c) the marginal PDF fy(y) is given by: fy(y) = ([tex]2\sqrt{(r^2 - y^2))/(\pi r^2}[/tex]), for -r < y < r

To find the marginal probability density functions (PDFs) fx(x) and fy(y) from the joint PDF fX,Y(x, y), we need to integrate the joint PDF over the appropriate range.

b) To find the marginal PDF fx(x), we integrate fX,Y(x, y) with respect to y while considering the range of x:

fx(x) = ∫fX,Y(x, y) dy

Since the joint PDF is defined over the circle [tex]x^2 + y^2 < r^2[/tex], the integration limits for y will be -[tex]\sqrt{r^2 - x^2)}[/tex] to [tex]\sqrt{r^2 - x^2)}[/tex]

fx(x) = ∫[tex][-\sqrt{(r^2 - x^2)}[/tex], [tex]\sqrt{(r^2 - x^2}[/tex])] (1/(πr^2)) dy

Integrating, we get:

fx(x) = ([tex]1/(\pi r^2)[/tex]) * 2[tex]\sqrt{(r^2 - x^2}[/tex]

Therefore, the marginal PDF fx(x) is given by:

fx(x) = (2[tex]\sqrt{(r^2 - x^2}[/tex]/([tex]\pi r^2[/tex]), for -r < x < r

c) Similarly, to find the marginal PDF fy(y), we integrate fX,Y(x, y) with respect to x while considering the range of y:

fy(y) = ∫fX,Y(x, y) dx

Since the joint PDF is defined over the circle[tex]x^2 + y^2 < r^2[/tex], the integration limits for x will be -[tex]\sqrt{r^2 - y^2}[/tex]) to [tex]\sqrt{(r^2 - y^2}[/tex]).

fy(y) = ∫[-[tex]\sqrt{r^2 - y^2}[/tex], √[tex]\sqrt{r^2 - y^2}[/tex]] ([tex]1/(\pi r^2[/tex])) dx

Integrating, we get:

fy(y) = (1/[tex](\pi r^2[/tex])) * 2[tex]\sqrt{r^2 - y^2}[/tex])

Therefore, the marginal PDF fy(y) is given by:

fy(y) = [tex]2\sqrt{(r^2 - y^2))/(\pi r^2}[/tex] for -r < y < r

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