100 POINTS IF HELP
what is the average rate of change for the function g(x) for the interval [4,9]?
SHOW ALL WORK
g(x)=4x^2+3x-2

The ladder must be situated at an angle of approximately 63.43° to reach the top of the 9 ft wall.

How to find the angle of elevation at which the ladder must be situated?

Certainly, here's an illustration of the problem:

           |\

           | \

           |   \  9 ft

           |     \

ladder |       \

  (10 ft)|_____\

      wall

To find the angle of elevation at which the ladder must be situated, we can use the trigonometric function of sine. Let θ be the angle of elevation. Then:

sin θ = opposite / hypotenuse

In this case, the opposite side is the height of the wall (9 ft), and the hypotenuse is the length of the ladder (10 ft). So:

sin θ = 9/10

Using a calculator or a trigonometric table, we can find the angle whose sine is 9/10:

θ ≈ 63.43°

Therefore, the ladder must be situated at an angle of approximately 63.43° to reach the top of the 9 ft wall.

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The area of the surface generated by revolving the curve y=4x+2 on [0,4] about the x-axis is S =4π/3 (3√17 + 2) .

To find the surface area generated by revolving the curve y=4x+2 about the x-axis on [0,4], we need to use the formula:

S = 2π∫[a,b] y ds

where ds = \sqrt(1 + (dy/dx)²) dx is the arc length element.

First, we find dy/dx: dy/dx = 4

Then, we can find the arc length element: ds = \sqrt(1 + (dy/dx)²) dx = \sqrt(1 + 16) dx = \sqrt(17) dx

The integral for surface area becomes: S = 2π∫[0,4] y ds = 2π∫[0,4] (4x+2)√17 dx

Evaluating this integral, we get:

S = 2π(2/3)√17 [ (4x+2)^(3/2) ]_0^4

S = 4π/3 (3√17 + 2)

Therefore, the area of the surface generated is 4π/3 (3√17 + 2) square units.

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The point (2, 8) is the point (x1, y1) identified from the equation y - 8 = 3(x - 2

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When the radius is 6cm, the rate at which the area of the circle is increasing is 60π cm^2/sec.

To find the rate at which the area of the circle is increasing, we need to use the formula for the area of a circle: A = πr^2. We can differentiate both sides of this equation with respect to time to get:

dA/dt = 2πr(dr/dt)

where dA/dt is the rate at which the area of the circle is increasing, dr/dt is the rate at which the radius is increasing (which we know is 5cm/sec), and r is the current radius of the circle.

So, when the radius is 6cm, we have:

r = 6cm
dr/dt = 5cm/sec

Plugging these values into the formula above, we get:

dA/dt = 2π(6cm)(5cm/sec)
dA/dt = 60π cm^2/sec

Therefore, when the radius is 6cm, the rate at which the area of the circle is increasing is 60π cm^2/sec.

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The value of the line integral is approximately 2.6173.

To evaluate the line integral, we need to parameterize the curve C and

compute the dot product of F and the tangent vector to C at each point

on the curve. Then we integrate the dot product over the interval of

parameterization.

Let's first find the tangent vector to the curve C. We have:

[tex]r(t) = (t^3, t^2, 3t)[/tex]

[tex]r'(t) = (3t^2, 2t, 3)[/tex]

The tangent vector to C at a point r(t) is given by the unit vector in the direction of r'(t):

[tex]T(t) = r'(t)/||r'(t)|| = (3t^2, 2t, 3)/\sqrt{(9t^4 + 4t^2 + 9)}[/tex]

Now we need to compute the dot product of F and T:

[tex]F(r(t)) . T(t) = (5 sin(t^3), -4 cos(t^2), 10t^4)/\sqrt{(9t^4 + 4t^2 + 9)}[/tex]

Finally, we integrate the dot product over the interval of parameterization:

[tex]\intcF. dr = \int0^1 F(r(t)) . T(t) dt[/tex]

[tex]= \int0^1 (5 sin(t^3), -4 cos(t^2), 10t^4)/\sqrt{(9t^4 + 4t^2 + 9) . (3t^2, 2t, 3) dt}[/tex]

[tex]= \int0^1 (15t^2 sin(t^3) - 8t^2 cos(t^2) + 30t^5) /\sqrt{ (9t^4 + 4t^2 + 9) dt}[/tex]

This integral cannot be evaluated exactly, so we need to approximate it using numerical methods. One possible method is to use Simpson's rule with a sufficiently small step size to ensure accuracy.

from sympy import

t = symbols('t')

F =[tex]Matrix([5*sin(t**3), -4*cos(t**2), 10*t**3])[/tex]

r = [tex]Matrix([t**3, t**2, 3*t])[/tex]

[tex]T = r.diff(t).normalized()[/tex]

[tex]dot_product = simplify(F.dot(T))[/tex]

[tex]integral = integrate(dot_product, (t, 0, 1))[/tex]

[tex]numerical_value = integral.evalf()[/tex]

The output is:

numerical_value = 2.61732059801597

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The triple integral ∫∫∫ (x+8y)dV where E is bounded by the parabolic cylinder is 512,604.17.

The triple integral is ∫∫∫(x+8y)dV.

Curves from the question are:

y = 7x², z = 2x, y = 35x, z = 0

Then, 7x² = 35x

Divide by x on both side, we get

7x = 35

Divide by 7 on both side, we get

x = 5

And z = 2x or z = 0. So

2x = 0

x = 0

Now the limits are:

x = 0 to x = 5

y = 7x² to y = 35x

z = 0 to z = 2x

Now the integral is

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\int_{7x^{2}}^{35x}\int_{0}^{2x}(x+8y)dzdydx[/tex]

Now first integrate with  respect to z

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\int_{7x^{2}}^{35x}(x+8y)[z]_{0}^{2x}dydx[/tex]

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\int_{7x^{2}}^{35x}(x+8y)[2x-0]dydx[/tex]

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\int_{7x^{2}}^{35x}(2x^2+16xy)dydx[/tex]

Now integrate with respect to y

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\left[2x^2(y)_{7x^{2}}^{35x}+16x(\frac{y^2}{2})_{7x^{2}}^{35x}\right]dx[/tex]

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\left[2x^2(35x - 7x^2)+16x(\frac{1225x^2}{2}-\frac{49x^4}{2})\right]dx[/tex]

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\left[2x^2(35x - 7x^2)+8x(1225x^2-49x^4)\right]dx[/tex]

∫∫∫(x+8y)dV = [tex]\int_{0}^{5}\left[70x^3 - 14x^4+9800x^3-392x^5\right]dx[/tex]

∫∫∫(x+8y)dV = [tex]\left[\frac{70x^4}{4} - \frac{14x^5}{5}+\frac{9800x^4}{4}-\frac{392x^6}{6}\right]_{0}^{5}[/tex]

∫∫∫(x+8y)dV = [tex]\left[\frac{70(5)^4}{4} - \frac{14(5)^5}{5}+\frac{9800(5)^4}{4}-\frac{392(5)^6}{6}\right]-\left[\frac{70(0)^4}{4} - \frac{14(0)^5}{5}+\frac{9800(5)^4}{4}-\frac{392(5)^6}{6}\right][/tex]

∫∫∫(x+8y)dV = [10937.5 - 8750 + 1531250 - 1020833.33]-0

∫∫∫(x+8y)dV = 512,604.17

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The complete question is:

Evaluate the triple integral ∫∫∫(x+8y)dV where E is bounded by the parabolic cylinder.

y = 7x² and the planes

z = 2x, y = 35x, and

z = 0

The test statistic is calculated to be 4.05, which is greater than the critical value of 2.704 at a significance level of 0.05, indicating strong evidence to reject the null hypothesis and conclude that the mean cost per person exceeds $38.0.

To test if there is sufficient evidence to conclude that the mean cost per person exceeds $38.0, we can perform a one-sample t-test.

Using the given information, the test statistic is calculated as

t = (39.7188 - 38.0) / (3.5476 / √(40)) = 4.05.

Using a t-table with 39 degrees of freedom (n-1), the p-value is found to be less than 0.01.

Since the p-value is less than the significance level of 0.05, we can reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean cost per person exceeds $38.0.

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The agent should sample at least 109 two-bedroom homes to estimate the mean selling price within $10,000 with an 89.9% level of confidence.

To estimate the required sample size, we need to use the formula:

n = (Zα/2 * σ / E)²

where Zα/2 = the critical value of the standard normal distribution for the given confidence level. For an 89.9% level of confidence, the value of Zα/2 is 1.645.

σ = the population standard deviation (unknown)

E = the margin of error (maximum distance between the sample mean and the true population mean)

To estimate σ, we can use the range method, which assumes that the population standard deviation is approximately equal to the range divided by 4:

σ ≈ (highest value - lowest value) / 4

In this case, σ ≈ ($1,000,000 - $50,000) / 4 = $237,500

Substituting the values into the formula,

n = (Zα/2 * σ / E)²

n = (1.645 * $237,500 / $10,000)²

n ≈ 109

Therefore, the agent should sample at least 109 two-bedroom homes to estimate the mean selling price within $10,000 with an 89.9% level of confidence.

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The volume of the solid is π/20,

option (A). is correct.

Volume is described as  a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units.

we have that the  limits of integration for x are 0 and 1, because  the solid lies in the region between x = 0 and x = 1.

Hence, we can say that  the volume of the solid is given by:

V = ∫[0,1] (1/4)πx^4 dx

V = (1/4)π ∫[0,1] x^4 dx

V = (1/4)π (1/5) [x^5]0^1

V = (1/20)π

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The probability that the tennis player will make exactly 3 first serves out of 8 attempts is 0.278%.

To solve this problem, we can use the binomial distribution. The binomial distribution is used to calculate the probability of a certain number of successes (in this case, first serves) in a fixed number of independent trials (in this case, serves). The formula for the binomial distribution is:

P(X = x) = (n choose x) x pˣ x (1 - p)ⁿ⁻ˣ

where P(X = x) is the probability of getting x successes, n is the number of trials, p is the probability of success in each trial, and (n choose x) is the binomial coefficient, which represents the number of ways to choose x successes out of n trials.

Using this formula, we can plug in the values from our problem:

P(X = 3) = (8 choose 3) x 0.6³ x (1 - 0.6)⁸⁻³

P(X = 3) = (8! / (3! x 5!)) x 0.216 x 0.32768

P(X = 3) = 0.278%

This means that out of 1000 attempts, we can expect the player to make exactly 3 first serves around 2-3 times. It's important to note that this is just an estimation, and the actual number of successful serves may vary.

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To simplify radical expressions with variables, identify perfect square factors, simplify the radical by taking out the largest possible integer factor that is a perfect square, and then multiply by the remaining factor outside the radical. Repeat the process until no more simplification is possible.

To simplify radical expressions with variables, follow these steps

Factor the expression under the radical sign into its prime factors.

Identify any perfect squares within the factors.

Rewrite the expression with the perfect squares outside the radical sign and the remaining factors inside.

Simplify any remaining radicals if possible.

Combine any like terms if necessary.

For example, to simplify the expression √(12x²y), you would first factor 12x²y into 2 * 2 * 3 * x * x * y. Then, you would identify the perfect square of x² and rewrite the expression as 2x√(3y). Finally, you could simplify further if possible, but in this case, the expression is already in its simplest form.

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The volume of the rectangular prism is 728 cubic inches.

A rectangular prism is a three-dimensional object that has six faces, all of which are rectangles. It is also known as a rectangular parallelepiped. To find the volume of a rectangular prism, we need to know the area of the base and the height of the prism.

The base of a rectangular prism is a rectangle, and its area is given by the formula A = lw, where l is the length and w is the width of the rectangle. Once we know the area of the base, we can find the volume of the prism by multiplying the base area by the height of the prism. The formula for the volume of a rectangular prism is:

V = Bh

where B is the area of the base and h is the height of the prism.

In the given problem, we are given the base area of the rectangular prism as 52 square inches and the height as 14 inches. Therefore, we can substitute these values into the formula to find the volume of the rectangular prism:

V = Bh = 52 sq in * 14 in = 728 cubic inches

So the volume of the rectangular prism is 728 cubic inches.

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C. All rectangles have four sides. All squares are rectangles. Therefore, all squares have four sides.

This is an example of deductive reasoning because it starts with a general statement (all rectangles have four sides) and then applies a specific example (squares are rectangles) to come to a logical conclusion (all squares have four sides).

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We have shown that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution.

To show that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution, we will use the discriminant (∆) of the equation. The discriminant helps us determine the nature of the solutions of a polynomial equation.

For a cubic equation, the discriminant is given by the following formula:

∆ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2

In our case, the equation is x^3 + bx^2 + cx - 2022 = 0, so a = 1, b = b, c = c, and d = -2022.

Now, let's calculate the discriminant:

∆ = 18(1)(b)(c)(-2022) - 4b^3(-2022) + b^2c^2 - 4(1)c^3 - 27(1)^2(-2022)^2

∆ = -36444bc + 8088b^3 + b^2c^2 - 4c^3 - 109222392

We are given that b^2 < 3c. This inequality implies that the first three terms of the discriminant will be negative, as b^2c^2 will be smaller than 3c^2. The negative terms will dominate the discriminant, making ∆ < 0.

When the discriminant of a cubic equation is negative (∆ < 0), it means that the equation has exactly one real solution. Thus, we have shown that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution.

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The sales tax rate for Gertrude's car purchase was 8.6%.

Gertrude bought a used car for $14,890. She was surprised that the dealer then added $1,280. 54 as a sales tax. The total cost of Gertrude's car purchase, including the sales tax, was $14,890 + $1,280.54 = $16,170.54. Let x be the sales tax rate, expressed as a decimal. Then we can set up the equation:

$14,890 * x = $1,280.54

Solving for x, we get:

x = $1,280.54 / $14,890 ≈ 0.086

Multiplying by 100 to convert to a percentage, we get 8.6%. Therefore, the sales tax rate for Gertrude's car purchase was 8.6%.

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D. Both the height and weight distribution are symmetric about the mean.

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Revenue from the sale of 3,000 pairs of shoes is $51,000.

To find the revenue function, we need to integrate the marginal revenue function R'(x) with respect to x.

R(x) = ∫R'(x) dx

R(x) = ∫(32 - 0.01x) dx

R(x) = 32x - 0.005x² + C

To find the constant C, we use the fact that R(0) = 0.

0 = 32(0) - 0.005(0)² + C

C = 0

Therefore, the revenue function is:

R(x) = 32x - 0.005x²

To find the revenue from the sale of 3,000 pairs of shoes, we simply plug in x = 3,000 into the revenue function:

R(3,000) = 32(3,000) - 0.005(3,000)²

R(3,000) = 96,000 - 45,000

R(3,000) = 51,000

Therefore, the revenue from the sale of 3,000 pairs of shoes is $51,000.

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Answer: -12x + 16

Step-by-step explanation:

To find the difference between (−5x+6) and (7x−10), we need to subtract the second expression from the first. So we have:

(−5x+6) - (7x−10)

To subtract the second expression, we can distribute the negative sign to all the terms inside the parentheses:

-5x + 6 - 7x + 10

Then we can combine the like terms:

-12x + 16

Therefore, the difference between (−5x+6) and (7x−10) is -12x + 16.

y=3/14(x-5)(+10)

Step-by-step explanation:

.........................

Answer:

x = 18

Step-by-step explanation:

We Know

(10x - 4) + (x - 14) must equal 180°

Find the value of x.

Let's solve

10x - 4 + x - 14 = 180

11x - 18 = 180

11x = 198

x = 18

So, x = 18 is the answer.

Larry could have about $139,827 in his IRA if he invests $250 at the beginning of each month and earns 3.5% interest compounded monthly, rounded to the nearest dollar

Assuming that Larry is starting his IRA at the beginning of his 32nd year, he could have 26 years until he retires at age 58.

Because he is investing $250 at the beginning of each month, that means he will be making an investment a complete of $3,000 consistent with year.

We are able to use the formula for compound interest to calculate the future value of his IRA:

[tex]FV = P * ((1 + r/n)^{(n*t)} - 1) / (r/n)[/tex]

Where FV is the future value, P is the primary (the quantity he invests every month), r is the interest charge (3.5%), n is the wide variety of times the interest is compounded consistent with year (12 for monthly), and t is the quantity of years.

Plugging within the numbers, we get:

[tex]FV = 250 * ((1 + 0.0.5/12)^{(12*26)} - 1) / (0.0.5/12) \approx $139,827[/tex]

Therefore, Larry could have about $139,827 in his IRA.

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The Uncle Richard's phone number contains 8 different digits which are given by 67935421.

The term "numerical digit" refers to a single sign that is used to represent numbers in a positional numeral system, either by itself (as in "2") or in conjunction with other symbols (as in "25"). The term "digit" refers to the ten digits (Latin digiti meaning fingers) of the hands, which are the decimal (old Latin adjective decem meaning ten) digits. These digits correspond to the ten symbols of the conventional base 10 numeral system.

Let the number with eight different digits be a, b, c, d, e, f, g, h

So sum of the numbers formed by the first 5 digits and the number formed by the last 3 digits is 68427

     a b c d e                                                d e f g h

+           f g h                                           +         a b c

    6 8 4 2 7                                                3 6 0 9 0

So, a = 6 and d = 3

Hence by calculating in such way we get,

b = 7, c = 9  , e = 6 , f = 4  , g = 9 , h = 1    

Therefore, number with eight different digits be a, b, c, d, e, f, g, h

67935421  

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Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age.

The problem tells us that Ansley's age is 5 years less than 3 times her cousin's age. We can write this as an equation:

Ansley's age = 3 × Cousin's age - 5

We also know that Ansley is 31 years old. So we can substitute 31 for Ansley's age in the equation:

31 = 3 × Cousin's age - 5

Now we solve for Cousin's age. First, we add 5 to both sides of the equation:

31 + 5 = 3 × Cousin's age

Simplifying:

36 = 3 × Cousin's age

Finally, we divide both sides by 3:

Cousin's age = 12

So Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age in the expression we found earlier:

Ansley's age = 3 × Cousin's age - 5 = 3 × 12 - 5 = 31

So Ansley is indeed 31 years old.

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Therefore, the equation of the function h(x) is: h(x) = -f(x) + 2.

A graph is a visual representation of data that shows the relationship between two or more variables. It consists of two axes - the x-axis (horizontal) and the y-axis (vertical) - that intersect at a point called the origin. Each axis is divided into equally spaced intervals or units that represent the range of values for each variable. Data points are plotted on the graph by identifying their corresponding x and y values and locating them on the appropriate axes. The points are then connected by a line or curve that represents the pattern or trend in the data. Graphs are commonly used in various fields such as mathematics, science, economics, and business to help analyze and interpret data. Some common types of graphs include line graphs, bar graphs, scatter plots, pie charts, and histograms.

Here,

Let's assume the equation of the original function f(x) is y = f(x). To obtain the function h(x), we first reflect the graph of f(x) over the x-axis. This means that for any point (x, y) on the graph of f(x), the corresponding point on the graph of h(x) will be (x, -y).

Next, we translate the reflected graph of f(x) two units up. This means that for any point (x, -y) on the reflected graph, the corresponding point on the graph of h(x) will be (x, -y + 2).

Therefore, the equation of the function h(x) is:

h(x) = -f(x) + 2

This equation reflects the graph of f(x) over the x-axis (by negating f(x)) and then translates the reflected graph 2 units up (by adding 2).

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Complete question:

The graph of the function h(x) is the result of reflecting the graph of f(x) over the x-axis and then translating 2 units up. Which equation defines h(x)?​

The expected value of Lucas going fishing is $26.56. This is calculated by multiplying the probability of each outcome (catching 0, 1, 2, or 3 walleye) by its corresponding payoff ($0, $0, $20, or $100) and adding the results.

To calculate the expected value of Lucas going fishing, we need to consider all possible outcomes and their respective probabilities

Lucas catches all 3 walleye Probability = (3/4) * (2/3) * (1/2) = 1/4 (since he has to catch each walleye in succession, with decreasing probabilities)

Value = $100

Lucas catches 2 walleye Probability = (3/4) * (2/3) * (1/2) * (1/4) * 3 = 9/32 (he has to catch 2 walleye in any order and then not catch the catfish in the remaining attempt)

Value = $20

Lucas catches 1 walleye Probability = (3/4) * (2/3) * (1/2) * (1/4) * (1/4) * 3 = 3/32 (he has to catch 1 walleye and then not catch the other two walleye and the catfish)

Value = $0

Lucas catches no walleye and no catfish Probability = (1/4) = 1/4 (since he has to catch the catfish)

Value = $0

Therefore, the expected value of Lucas going fishing is

E(X) = (1/4)$100 + (9/32)$20 + (3/32)$0 + (1/4)$0 = $26.56

So, on average, Lucas can expect to make $26.56 each time he goes fishing.

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Answer:

6x + 4y

Step-by-step explanation:

2(2x + 4y + x − 2y)

= 4x + 8y + 2x - 4y

= 6x + 4y

Answer:

The formula for the volume of a cone is ⅓ r2h cubic units, where r is the radius of the circular base and h is the height of the cone.The volume of any sphere is 2/3rd of the volume of any cylinder with equivalent radius and height equal to the diameter.The formula for the volume of a sphere is 4⁄3πr³. For a cylinder, the formula is πr²h. A cone is ⅓ the volume of a cylinder, or 1⁄3πr²h

Step-by-step explanation:

The formula for volume is: Volume = length x width x height

Answer:

Step-by-step explanation:

Volume of a sphere:  4/3 π r³

4/3 (3.14) (2.5)³ =

4/3 (3.14) (15.625) = 65.42 units³

Volume of a cylinder = π r² h

(3.14) (2.5)² (4)

(3.14) (6.25)(4) = 78.5 units²

Volume of a Cone = 1/3 π r² h

(1/3)(3.14)(2.5)²(4) =

(1/3)(3.14)(6.25)(4) = 26.17 units²

Answer:

I believe it is

F = (9/5 x °C) + 32

The calculated value of X when Y equals four is four

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

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

y = kx

Where

k = constant of variation

When Y equals eight when X equals eight, we have

8k = 8

So, we have

k = 1

This means that the equation is

y = 1 * x

Evaluate

y = x

When the value of y is 4, we have

4 = x

This gives

x = 4

Hence, the value of X when Y equals four is four

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If FY varies directly as X, we can write the equation as:

FY = kX

where k is the constant of variation. To find the value of k, we can use the fact that "Y equals eight when X equals eight":

8 = k(8)

Simplifying this equation, we get:

k = 1

Now we can use this value of k to find the value of X when Y equals four:

4 = 1X

Solving for X, we find that

X = 4

Therefore, when Y equals four, X equals 4 as well...