A ball is thrown into the air with an initial velocity of 30 ft/sec. This situation is modeled by h=-16t^2+30t+6. Will it reach the top of a building with a roof height of 22 feet?

You can be​ 90% confident that the population mean price of the stock is between the bounds of the​ 90% confidence​ interval, and​ 95% confident for the​ 95% interval.

Given data ,

The problem states that a random sample of 43 business days was taken, and the mean closing price of the stock in that sample was $112.15. The population standard deviation is assumed to be $9.95. Based on this information, a confidence interval can be calculated for the population mean.

Now , A wider interval results from a greater confidence level since it calls for more assurance.

If you compare the offered alternatives, option C accurately indicates that you can have a 90% confidence interval for the population mean price of the stock being inside the boundaries, and a 95% confidence interval. Because a 90% confidence interval demands more assurance than a 95% confidence interval, it is smaller. As a result, the population mean is more likely to fall inside the 90% confidence interval's boundaries.

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The solution to Ax=b using the LU factorization is: x = [27 -23/4 -30]

To find the solution to the system Ax=b using the LU factorization:

We need to first decompose the matrix A into its lower and upper triangular matrices L and U respectively, such that A = LU.

Using the given LU factorization of A, we can write:

L = [1 0 0] [1 0 0] [-1 3 1]

U = [2 3 4] [0 -4 3] [0 0 -1]

Next, we need to solve the system LY=b. We can substitute L and Y with their corresponding matrices and variables respectively:

[1 0 0] [1 0 0] [-1 3 1] [y1 y2 y3] = [4 17 43]

Simplifying this system, we get:

y1 = 4

y2 = 17

-y1 + 3y2 + y3 = 43

Solving for y3, we get:

y3 = 30

Now that we have the values for Y, we can solve the system Ux=Y to get the solution to Ax=b.

We can substitute U and X with their corresponding matrices and variables respectively:

[2 3 4] [0 -4 3] [0 0 -1] [x1 x2 x3] = [y1 y2 y3]

Simplifying this system, we get:

2x1 + 3x2 + 4x3 = 4

-4x2 + 3x3 = 17

-x3 = 30

Solving for x3, we get:

x3 = -30

Substituting x3 into the second equation, we get:

-4x2 + 3(-30) = 17

Solving for x2, we get:

x2 = -23/4

Substituting x2 and x3 into the first equation, we get:

2x1 + 3(-23/4) + 4(-30) = 4

Solving for x1, we get:

x1 = 27

Therefore, the solution to Ax=b using the LU factorization is:

x = [27 -23/4 -30]

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Answer:

The pressure exerted by the box on the shelf is given by the formula:

Pressure = Force / Area

where Pressure is measured in Newtons per square meter (N/m^2), Force is measured in Newtons (N), and Area is measured in square meters (m^2).

We are given that the pressure exerted by the box on the shelf is 200 N/m and the force that the box exerts on the shelf is 140 N. Using the formula above, we can solve for the area of the base of the box as follows:

200 N/m = 140 N / Area

Simplifying the equation above, we can multiply both sides by the Area to get:

Area * 200 N/m = 140 N

Dividing both sides by 200 N/m, we get:

Area = 140 N / 200 N/m

Simplifying the right-hand side, we get:

Area = 0.7 m^2

Therefore, the area of the base of the box is 0.7 square meters, or 0.7 m^2 to 1 decimal place.

Answer:

0.7 m²

Step-by-step explanation:

The pressure exerted by the box on the shelf is defined as the force per unit area, so we can use the formula:

[tex]\boxed{\sf Pressure = \dfrac{Force}{Area}}[/tex]

We need to determine the area of the base of the box, so we can rearrange the formula to solve for area:

[tex]\boxed{\sf Area= \dfrac{Force}{Pressure}}[/tex]

Given values:

Substitute the given values into the formula:

[tex]\implies \sf Area = \dfrac{140\;N}{200\;N\;m^{-2}}[/tex]

[tex]\implies \sf Area = \dfrac{140}{200}\;m^2[/tex]

[tex]\implies \sf Area = 0.7\;m^2[/tex]

Therefore, the area of the base of the box is 0.7 square meters.

According to the information, we can find that the t-value for the lower endpoint is approximately -1.684, and the t-value for the upper endpoint is approximately 1.684.

To find the t-values for each of the given cases, we can use a t-distribution table or a calculator. Here are the answers for each case:

A) Upper tail area of .025 with 12 degrees of freedom:

The t-value for an upper tail area of .025 with 12 degrees of freedom is approximately 2.179.

B) Lower tail area of .05 with 50 degrees of freedom:

The t-value for a lower tail area of .05 with 50 degrees of freedom is approximately -1.677.

C) Upper tail area of .01 with 30 degrees of freedom:

The t-value for an upper tail area of .01 with 30 degrees of freedom is approximately 2.750.

D) Where 90% of the area falls between these two t values with 25 degrees of freedom:

We need to find the t-values that correspond to the middle 90% of the t-distribution with 25 degrees of freedom. This means that we want to find the t-values that divide the area under the curve into two equal parts, each with 45% of the area.

Using a t-distribution table or a calculator, we can find that the t-value for the lower endpoint is approximately -1.708, and the t-value for the upper endpoint is approximately 1.708.

E) Where 95% of the area falls between these two t values with 45 degrees of freedom:

We need to find the t-values that correspond to the middle 95% of the t-distribution with 45 degrees of freedom. This means that we want to find the t-values that divide the area under the curve into two equal parts, each with 2.5% of the area.

Using a t-distribution table or a calculator, we can find that the t-value for the lower endpoint is approximately -1.684, and the t-value for the upper endpoint is approximately 1.684.

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Answer: P≈15.5 units.

Step-by-step explanation:

The perimeter of a triangle is equal to the sum of all its sides:

                                          P = a + b + c,

where P is the perimeter and a, b, c are the sides of the triangle.

The segment length formula makes it possible to calculate the distance between two arbitrary points in the plane, provided that the coordinates of these points are known:

                               [tex]\boxed {d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2} }[/tex]

1) (1,6)   (3,1)  ⇒   x₁=1    x₂=3     y₁=6      y₂=1

[tex]a=\sqrt{(1-3)^2+(6-1)^2} \\\\a=\sqrt{(-2)^2+5^2} \\\\a=\sqrt{4+25} \\\\a=\sqrt{29} \approx5.4\ units\\[/tex]

2) (1,6)   (6,1)  ⇒  x₁=1    x₂=6   y₁=6   y₂=1

[tex]b=\sqrt{(1-6)^2+(6-1)^2} \\\\b=\sqrt{(-5)^2+5^2} \\\\b=\sqrt{25+25} \\\\b=\sqrt{50} \approx7.1\ units\\[/tex]

3) (3,1)   (6,1)   ⇒   x₁=3   x₂=6   y₁=1   y₂=1

[tex]c=\sqrt{(3-6)^2+(1-1)^2} \\\\c=\sqrt{(-3)^2+0^2} \\\\a=\sqrt{9+0} \\\\a=\sqrt{9} =3\ units\\[/tex]

4) P=a+b+c

P≈5.4+7.1+3

P≈15.5 units.

The value of c in the quadratic equation given is 32.

Given a quadratic equation of the form 3x² + bx + c = 0 has roots of 6 plus or minus square root of 2, we know that the quadratic equation can be written as:

3(x - (6 + √2))(x - (6 - √2)) = 0

Expanding this product gives:

3[(x - 6 - √2)(x - 6 + √2)] = 0

Using the difference of squares, we can simplify this expression to:

3[(x - 6)² - (√2)²] = 0

3(x - 6)² - 6 = 0

Multiplying out the squared term, we get:

3x² - 36x + 102 - 6 = 0

Simplifying, we get:

3x² - 36x + 96 = 0

Dividing both sides by 3, we get:

x² - 12x + 32 = 0

Therefore, the value of c is 32.

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The values of the missing sides and angles using trigonometric ratios are:

b = 7.06

c = 3.76

C = 28°

The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent.

The symbols used for them are:

sine: sin

cosine: cos

tangent: tan

cosecant: csc

secant: sec

cotangent: cot

The trigonometric ratios are defined as the ratio of the sides in right triangles.

Using trigonometric ratios, we have:

b/8 = sin 62

b = 8 * sin 62

b = 7.06

Similarly:

c/8 = cos 62

c = 8 * cos 62

c = 3.76

Sum of angles in a triangle is 180 degrees. Thus:

C = 180 - (90 + 62)

C = 28°

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The separation of internal and translational motion involves the reduced mass µ, which simplifies the motion of a two-particle system.

The reduced mass µ is calculated as µ = m₁m₂/(m₁ + m₂), and its inverse relationship is 1/µ = 1/m₁ + 1/m₂. The coordinates x1 and x2 are represented as x1 = X + m₂/mₓ and x2 = X - m₁/mₓ, respectively.

In a two-particle system, separating internal and translational motion allows us to simplify the analysis of the system's behavior. The reduced mass, µ, is a scalar quantity that effectively replaces the two individual masses, m₁ and m₂, in the equations of motion.

The coordinates x1 and x2 help to describe the positions of the particles in the system. By calculating the reduced mass and the coordinates x1 and x2, we can more easily examine the internal and translational motion of the particles and understand their interactions within the system.

This separation allows for more efficient problem-solving in the study of particle dynamics.

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Answer:

interior angle = 58°; exterior angle = 122°

Step-by-step explanation:

For all polygons, an interior angle and its accompanying exterior angle are always supplementary and thus equal 180°.

Thus, we can first find g by making the sum of the equation given for the interior angle and the equation given for the exterior angle equal to 180 and solve for g:

[tex](2g+16)+(4g+38)=180\\2g+16+4g+38=180\\6g+54=180\\6g=126\\g=21[/tex]

Now, we can first find the measure of interior angle X by plugging in g for 21:

[tex]X=2(21)+16\\X=42+16\\X=58[/tex]

Finally, we can find the measure of the exterior angle by either plugging in g for the equation or simply by subtracting 58 from 180 since the interior and exterior angle are supplementary and equal 180:

Exterior angle = 180 - 58

Exterior angle = 122

The slope of the parametric curve x=-[tex]2t^3[/tex]+7, y=3t² at the point corresponding to t is -1 divided by t.

To find the slope of the parametric curve x=-[tex]2t^3[/tex]+7, y=3t², we need to take the derivative of y with respect to x.

To do this, we can use the chain rule:

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

where (dx/dt) is the derivative of x with respect to t, and (dy/dt) is the derivative of y with respect to t.

Taking the derivatives, we get:

dx/dt = -6t²

dy/dt = 6t

Substituting these values, we get:

(dy/dx) = (dy/dt) / (dx/dt) = (6t) / (-6t²) = -1/t

So, the slope of the curve at the point corresponding to t is -1/t.

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You can use inequality signs to show lower and upper bounds of a number.

For example:

Lower bound:

x ≥ 5 (means x is greater than or equal to 5)

Upper bound:

x ≤ 10 (means x is less than or equal to 10)

Together they show a range:

5 ≤ x ≤ 10 (means x is between 5 and 10)

Some other examples:

0 < x < 100 (means x is between 0 and 100)

-10 ≤ y ≤ 50 (means y is between -10 and 50)

-5 < z < 12.5 (means z is between -5 and 12.5)

Does this help explain using inequalities to show boundaries or ranges of numbers? Let me know if you have any other questions!

The value of dy for y = [tex]e^{(x/10)}[/tex], x = 0, and dx = 0.1 is 0.01.

To evaluate the value of dy for the given values of x and dx, we first need to find the derivative of y with respect to x, which can be computed as follows:

[tex]y = e^{(x/10)}[/tex]

Differentiating both sides with respect to x using the chain rule, we get:

dy/dx = d/dx [[tex]y = e^{(x/10)}\\[/tex]]

=[tex]y = e^{x/10}[/tex] * d/dx [x/10]

= [tex]y = e^{(x/10)}[/tex] * (1/10) * d/dx [x]

=[tex]y = e^{(x/10)}[/tex] * (1/10)

Now, we substitute the values x = 0 and dx = 0.1 in the above expression to get the value of dy:

dy = (1/10) *[tex]e^{(0/10)}[/tex] * dx

= (1/10) * (1) * (0.1)

= 0.01

Therefore, the value of dy for y = [tex]e^{(x/10)}[/tex], x = 0, and dx = 0.1 is 0.01.

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As per given just by subtracting Maureen's book from Francesca's book. Maureen's book has 156 more pages than Maureen's book.

Subtraction is a mathematical operation that involves finding the difference between two numbers. It is one of the four basic arithmetic operations, along with addition, multiplication, and division.

Subtraction is used to determine how much more or less of one quantity there is compared to another quantity. For example, if you have 10 apples and you give away 3, then you have 7 apples left. The difference between the initial amount of apples (10) and the amount after giving away (7) is found through subtraction: 10 - 3 = 7.

To find out how many more pages Francesca's book has than Maureen's book, we can subtract the number of pages in Maureen's book from the number of pages in Francesca's book:

Francesca's book - Maureen's book = 434 - 278 = 156

Therefore, Francesca's book has 156 more pages than Maureen's book.

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Answer:

The formula for average cost (AC) is:

AC = (Total cost / Quantity)

To find the total cost, we need to add the variable cost (VC) and the avoidable fixed cost:

Total cost = VC + Fixed cost

Total cost = 20Q + 50,000

Now we can substitute this into the formula for average cost:

AC = (Total cost / Quantity)

AC = (20Q + 50,000) / Q

Simplifying this expression gives:

AC = 50,000/Q + 20

Therefore, the firm's average cost function is:

AC = 50,000/Q + 20

So, the correct answer is B.

The solution of the quadratic equation  4e² - 15e = -4 in the exact form is e = (15 + √161)/8 or e = (15 - √161)/8

To solve the quadratic equation 4e² - 15e = -4, we can rearrange it into standard form as follows,

4e² - 15e + 4 = 0. We can then use the quadratic formula, which states that for an equation in the form ax² + bx + c = 0, the solutions are given by,

x = (-b ± √(b² - 4ac)) / 2a

Applying this formula to our equation, we have,

e = (-(-15) ± √((-15)² - 4(4)(4))) / 2(4)

Simplifying this expression, we get,

e = (15 ± √(225 - 64)) / 8

e = (15 ± √161) / 8

Therefore, the solutions to the equation 4e² - 15e = -4 are:

e = (15 + √161) / 8 or e = (15 - √161) / 8

These are exact solutions in radical form.

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Given that x is a continuous variable with the probability density function f(x) = c(10-x)^2 for 0 < x < 10, we need to find the value of c.

Step 1: Understand that for a probability density function, the total area under the curve must equal 1. Mathematically, this is expressed as:

∫[f(x)] dx = 1, with integration limits from 0 to 10.

Step 2: Substitute f(x) with the given function and integrate:

∫[c(10-x)^2] dx from 0 to 10 = 1

Step 3: Perform the integration:

c ∫[(10-x)^2] dx from 0 to 10 = 1

Step 4: Apply the power rule for integration:

c[(10-x)^3 / -3] from 0 to 10 = 1

Step 5: Substitute the integration limits:

c[(-1000)/-3 - (0)/-3] = 1

Step 6: Solve for c:

(1000/3)c = 1

c = 3/1000

c = 0.003

So the probability density function f(x) = 0.003(10-x)^2 for 0 < x < 10.

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The decision complexity advantage for 10 decisions with two alternatives compared to one decision with 20 alternatives is 3.31. This can be answered by the concept of Log.

To calculate the decision complexity advantage, we need to first plug in the given values for a and b into the formula RT = a + b log2(N), where N is the number of alternatives.

For 10 decisions with two alternatives each, N = 2¹⁰ = 1024. Thus, RT = 1 + 2 log2(1024) = 22 seconds.

For one decision with 20 alternatives, N = 20. Thus, RT = 1 + 2 log2(20) = 6.64 seconds.

The decision complexity advantage is calculated by taking the ratio of the RT values: 22/6.64 = 3.31. This means that making 10 decisions with two alternatives each is 3.31 times faster than making one decision with 20 alternatives.

Therefore, the decision complexity advantage for 10 decisions with two alternatives compared to one decision with 20 alternatives is 3.31.

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The triangular prism has 3 cylindrical holes with a diameter of 4 cm. The volume of each hole is approximately 60π cubic centimeters, so the total volume of all three holes is about 180π cubic centimeters.

To find the volume of cylindrical holes in the triangular prism, we need to calculate the volume of one cylinder and then multiply it by three (since there are three cylindrical holes).

Volume of one cylinder = πr²h, where r is the radius of the cylinder and h is the height.

Given the diameter of the cylindrical hole is 4 cm, we can find the radius by dividing it by 2

radius (r) = 4 cm ÷ 2 = 2 cm

The height of the cylinder is the same as the length of the prism, which is 15 cm.

Volume of one cylinder = π(2 cm)² × 15 cm

= 60π cm³

Since there are three cylindrical holes, the total volume of the holes is

Total volume of cylindrical holes = 3 × 60π cm³

= 180π cm³

Therefore, the volume of the three cylindrical holes is 180π cubic centimeters.

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--The given question is incomplete, the complete question is given

" Pls help (part 1)

Find the volume of 3 cylindrical holes.

Give step by step explanation! "--

Answer:

Step-by-step explanation:

To solve this system of inequalities, we can first rearrange the first equation to solve for y:

2y = 3x - 16

y = (3/2)x - 8

Now we can substitute this expression for y into the second inequality:

y + 2x > -5

(3/2)x - 8 + 2x > -5

(7/2)x > 3

x > 6/7

So the solution to the system of inequalities is:

y > (-5 - 2x)

x > 6/7

Answer:

no solution

no absolute max or min

Step-by-step explanation:

The correct graph of equation on the coordinate plane is shown in figure.

We know that;

The equation of line with slope m and y intercept at point b is given as;

y = mx + b

Here, The equation is,

y = 2/3x + 1

Hence, Slope of equation is, 2/3

And, Y - intercept of the equation is, 1

Thus, The correct graph of equation on the coordinate plane is shown in figure.

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When the unit rate of the horse and the fox is compared, the statement that is true about them will be that The fox was 0.3 miles/minute faster than the horse. That is option C.

From the graph,

30 miles distance covered by the horse = 60 mins

But 60 mins = 1 hours

Therefore, the rate of distance covered by the horse = 30 miles/hr.

But the rate of distance covered in miles/ min = 5/10 = 0.5 miles/min.

If the fox covers 0.8miles/min then the difference between it and the horse = 0.8-0.5 = 0.3miles/min.

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Your answer: The elasticity of the demand function 2p 3q = 90 at the price p = 15 is -0.5.

To find the elasticity of the demand function, we need to use the following formula:

Elasticity = (dq/dp) * (p/q)

where dq/dp is the derivative of q with respect to p, and (p/q) is the ratio of the two variables at a given point.

First, we need to solve the demand function for q in terms of p:

2p + 3q = 90

3q = 90 - 2p

q = (90 - 2p)/3

Next, we need to find the derivative of q with respect to p:

dq/dp = (-2/3)

Finally, we can plug in the values for p and q to find the elasticity at p = 15:

q = (90 - 2(15))/3 = 20

(p/q) = 15/20 = 0.75

Elasticity = (-2/3) * (15/20) = -0.5

Therefore, the elasticity of the demand function 2p + 3q = 90 at the price p = 15 is -0.5. This means that a 1% increase in price would lead to a 0.5% decrease in quantity demanded.

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The correct answer is: It is necessary to apply both the decomposition and the augmentation rules to F in order to form a minimal cover.

To form a minimal cover of a set of functional dependencies, we need to apply the decomposition rule, which involves breaking down each dependency in F into its simplest form, and the augmentation rule, which involves adding any missing attributes to the right-hand side of each dependency. In this case, we need to apply both rules to F to obtain a minimal cover.

For example, applying the decomposition rule to UVX->UW yields two dependencies: UV->UW and UX->UW. Applying the augmentation rule to UX->ZV yields UX->ZVY. Continuing in this way, we can obtain a minimal cover for F, which is:

UV->UW
UX->ZVY
VU->Y
V->Y
W->VY
a. It is necessary and sufficient to remove a dependency W->Y from F to form a minimal cover.

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Answer:

1. -10 is a coefficient

2. B

3. C

4. B

5. 29.6

6. n=8

7. ?

8. C

Answer:

perimeter = 22

area = 26

All four requirements are satisfied and the given procedure does result in a binomial distribution. The answer is OC, "Yes, because all 4 requirements are satisfied."

The given procedure does result in a binomial distribution. The four requirements for a binomial distribution are:
1) The experiment consists of a fixed number of trials.
2) Each trial has only two possible outcomes, success or failure.
3) The trials are independent of each other.
4) The probability of success remains the same for each trial.

In this case, each mp3 player can either be acceptable or defective, so there are only two possible outcomes. The trials are independent of each other, and the probability of a player being acceptable or defective remains the same for each trial.

Therefore, all four requirements are satisfied and the given procedure does result in a binomial distribution. The answer is OC, "Yes, because all 4 requirements are satisfied."

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a. The instantaneous rate of growth can be found by taking the derivative of the growth model Eq. (5.18) with respect to time.

b. The compound rate of growth can be calculated by using the formula: [(1+instantaneous rate of growth)ⁿ]-1, where n is the number of periods.

c. The difference in the two slope coefficients for the S&P data may be due to changes in the underlying economic conditions or external factors affecting the market. To reconcile the difference, a more detailed analysis should be conducted to identify the specific factors contributing to the change in slope coefficients.

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S ≈ -0.0010 (rounded to four decimal places).

To approximate the value of the series ∑n=1infinityn / (n^2)(n^6) within an error of at most 10^(-3), we can use the alternating series test and the remainder formula.

The series is alternating because the sign alternates between positive and negative. Moreover, the terms of the series are decreasing in absolute value because:

|(-1)^(n+1) / (n^2)(n^6)| < |(-1)^(n) / ((n+1)^2)((n+1)^6)| for all n

Therefore, we can apply the alternating series test and bound the error by the absolute value of the first neglected term:

|R_N| = |-1^(N+1) / (N+1)^2((N+1)^6)|

To find the smallest value of N that approximates S to within an error of at most 10^(-5), we need to solve the inequality:

|R_N| = |-1^(N+1) / (N+1)^2((N+1)^6)| ≤ 10^(-5)

Solving for N, we get:

N ≥ 14

Thus, the smallest value of N that approximates S to within an error of at most 10^(-5) is N=14.

To approximate S, we can sum the first 14 terms of the series:

S ≈ ∑n=114^n / (n^2)(n^6)

Using a calculator or a computer algebra system, we get:

S ≈ -0.00102583...

Therefore, S ≈ -0.0010 (rounded to four decimal places).

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To calculate E(X|Y), we need to find the conditional PDF of X given Y. Using the given joint PDF, we can find the conditional PDF as

  f(X|Y) = (6XY) / (3Y^2) = 2X / Y for 0 ≤ X ≤ Y.

Then, we can find the conditional expectation as

E(X|Y) = ∫X f(X|Y) dX, which evaluates to

E(X|Y) = 2/3 Y²

   2. Calculate Var(X|Y):

To calculate Var(X|Y), we need to first find the conditional expectation of X given Y, which we calculated in the previous step as

  E(X|Y) = 2/3 Y².

Then, we can find the conditional variance of X given Y as

  Var(X|Y) = E(X²|Y) - [E(X|Y)]²,

 where E(X²|Y) = ∫X² f(X|Y) dX.

After computing the integrals, we get

  Var(X|Y) = (2/5)[tex]Y^3[/tex] - (4/9)[tex]Y^4[/tex]

     3. Show that E[E(X|Y)] = E(X):

We can show that E[E(X|Y)] = E(X) using the "Conditional Probability" , which states that E(X) = E[E(X|Y)].

From the previous calculations, we know that E(X|Y) = 2/3 Y², and the marginal PDF of Y is f(Y) = 3Y² for 0 ≤ Y ≤ 1.

Therefore, we can compute E(E(X|Y)) as E(E(X|Y)) = ∫Y E(X|Y) f(Y) dY, which evaluates to E(E(X|Y)) = 2/5.

Also, we previously computed E(X) as E(X) = 3/2.

Therefore, we have E[E(X|Y)] = 2/5 and E(X) = 3/2, and

we can see that E[E(X|Y)] ≠ E(X).

This indicates that X and Y are dependent variables.

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The minimum possible value of f ( 7 ) − f ( 3 ) f(7)-f(3) is −4 and the maximum possible value is 4.

Given that 2 ≤ f ' ( x ) ≤ 4 2≤f′(x)≤4 for all values of x, we can make use of the Mean Value Theorem to determine the minimum and maximum possible values of f ( 7 ) − f ( 3 ) f(7)-f(3).

According to the Mean Value Theorem, there exists a c ∈ ( 3 , 7 ) c\in(3,7) such that:

f ( 7 ) − f ( 3 ) = f ′ ( c ) ( 7 − 3 ) = 4 c − 12 4c-12

Since f'(x) is between 2 and 4 for all values of x, we know that 8 ≤ 4c ≤ 16 8\leq4c\leq16. Therefore, 2 ≤ c ≤ 4 2\leq c\leq 4.

To find the maximum value of f ( 7 ) − f ( 3 ) f(7)-f(3), we need to maximize 4c-12 when c is between 2 and 4. This occurs when c = 4, so the maximum value of f ( 7 ) − f ( 3 ) f(7)-f(3) is:

f ( 7 ) − f ( 3 ) ≤ 4 ( 4 ) − 12 = 4

To find the minimum value of f ( 7 ) − f ( 3 ) f(7)-f(3), we need to minimize 4c-12 when c is between 2 and 4. This occurs when c = 2, so the minimum value of f ( 7 ) − f ( 3 ) f(7)-f(3) is:

f ( 7 ) − f ( 3 ) ≥ 4 ( 2 ) − 12 = −4

Therefore, the minimum possible value of f ( 7 ) − f ( 3 ) f(7)-f(3) is −4 and the maximum possible value is 4.

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