Question number 5. What is the relation called

AABC is transformed onto ADEF by a dilation by a scale factor of 2, centered at the origin, followed by a reflection over the y-axis.

The sequence of transformations that will move AABC onto ADEF is a dilation by a scale factor of 2, centered at the origin, followed by a reflection over the y-axis.

Firstly, dilation is a transformation that changes the size of an object but not its shape.

The dilation factor is multiplied by each coordinate, so when the dilation is centered at the origin, the new coordinates will be twice the original coordinates.

Therefore, AABC will be enlarged to A'BC', and DEF will be enlarged to D'E'F, both with double the size.

Then, reflection is a transformation that flips an object over a line of reflection. In this case, the line of reflection is the y-axis.

When we reflect A'BC' over the y-axis, we get A''B''C'', and when we reflect D'E'F over the y-axis, we get D''E''F''.

Therefore, AABC is transformed onto ADEF by a dilation by a scale factor of 2, centered at the origin, followed by a reflection over the y-axis.

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To construct a square and a regular pentagon with equal side lengths of 0.5 inch, we need to use basic geometric constructions.

To construct a square and a regular pentagon with equal side length of 0.5 inch, follow these steps:

(a) Construct a Square:

Connect the endpoints of the four line segments to form a square.

(b) Construct a Regular Pentagon:

Repeat this process for all five vertices of the circle to form the regular pentagon.

A geometric construction is a method of drawing a figure using only a straightedge (an unmarked ruler) and a compass.

For the square, we start by drawing a horizontal line segment of length 0.5 inch. We then draw two perpendicular lines of length 0.5 inch each, meeting at the endpoints of the original line segment.

These two new line segments represent the adjacent sides of the square. We then repeat this process to create the remaining two sides of the square, and connect all four endpoints to form the complete square.

For the regular pentagon, we need to construct a circle with a radius of 0.5 inch. This will be the circumcircle of the pentagon, meaning that all five vertices of the pentagon will lie on the circle.

We draw a horizontal line through the center of the circle, and mark the points where the line intersects the circle. These five points will be the vertices of the pentagon.

We then draw line segments connecting adjacent vertices, using a compass to copy the length of each line segment from the previous one.

This process will create all five sides of the pentagon, and the figure will be a regular pentagon with equal side lengths of 0.5 inch.

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Given:- ABCD is a parallelogram, and AE and CF bisect ∠A and ∠C respectively. To prove:- AE∥CF Proof:- Since in a parallelogram, opposite angles are equal.

A parallelogram is a geometric shape that has four sides and four angles. It is a type of quadrilateral, which means it has four sides, and its opposite sides are parallel to each other.

The opposite sides of a parallelogram are also equal in length. The opposite angles of a parallelogram are also equal in measure.

The shape of a parallelogram looks similar to a rectangle, but it differs from a rectangle in that its angles are not necessarily right angles. A square is a special case of a parallelogram in which all four sides are equal in length and all four angles are right angles

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The recursive sequence representing how many papers Khloe has remaining to grade after working for n hours is given by a_n = a_{n-1} - 10, where a_0 = 90.

Let a_n denote the number of papers Khloe has remaining to grade after n hours of work. After the first hour of work, she will have 90 - 10 = 80 papers remaining. Therefore, we have a_1 = 90 - 10 = 80.

After the second hour of work, she will have a_2 = a_1 - 10 = 80 - 10 = 70 papers remaining. Similarly, after the third hour of work, she will have a_3 = a_2 - 10 = 70 - 10 = 60 papers remaining.

In general, after n hours of work, Khloe will have a_n = a_{n-1} - 10 papers remaining to grade. This is a recursive sequence, where the value of a_n depends on the value of a_{n-1}. The initial value of a_0 is given as 90, since she starts with 90 papers to grade. Therefore, the recursive sequence is given by a_n = a_{n-1} - 10, where a_0 = 90.

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The volume of the campfire pyramid is approximately 8,175 cubic centimeters. This is found by using the formula for the volume of a pyramid and plugging in the given values for the height and base area of the pentagonal pyramid.

The formula for the volume of a pyramid is

V = (1/3) * base_area * height

where V is the volume, base_area is the area of the base, and height is the height of the pyramid.

In this case, the height of the pyramid is given as 27 cm, and the base area is given as 965 square cm. To find the volume, we can plug these values into the formula

V = (1/3) * 965 cm² * 27 cm

V = 8,175 cm³

Therefore, the volume of the campfire pyramid is 8,175 cubic centimeters.

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The total number of customers that the waiter had would be = 30 customers.

The total number of tables the waiter had = 5 tables

The total number of women at each table = 3

The total number of men at each table = 3

The total number of people one each table = 6

Therefore the total number of customers that the waiter attended to would be = 5×6 = 30

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The potters are qualified for the home loan as their total monthly payment is $831.02, which is less than $1000.00.

The total cost of the cottage along with the annual insurance and taxes is $118,000 + $710 + $2800 = $121,510.

The down payment made by the potters is $14,000. Therefore, the amount to be financed through a mortgage is $121,510 - $14,000 = $107,510.

Using the formula for the monthly payment of a mortgage, which is given by:

M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1]

where P is the principal (amount to be financed), i is the monthly interest rate, and n is the total number of monthly payments.

For a 5%, 15-year mortgage, the monthly interest rate is 0.05/12 = 0.0041667, and the total number of monthly payments is 15 x 12 = 180.

Plugging in the values, we get:

M = $107,510 [ 0.0041667 (1 + 0.0041667)^180 ] / [ (1 + 0.0041667)^180 - 1 ]

M = $831.02

Therefore, the total monthly payment for the mortgage and the annual insurance and taxes is $831.02 + $59.17 + $233.33 = $1123.52, which is more than the maximum allowed payment of $1000.00. Hence, the potters are qualified for the home loan.

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The width should be 12 feet and the height should be 6 feet.

Let's assume that the rectangular part of the window has a width of x feet. Since the semi-circle at the top is half the width of the rectangle, its radius will also be x/2 feet. Therefore, the height of the rectangle can be expressed as 12 - x/2, since we have a total of 12 feet of frame material available.

To find the area of the rectangle, we can multiply its length and width together: A = x(12 - x/2) = 12x - x²/2. To maximize this area, we can take its derivative with respect to x and set it equal to 0:

dA/dx = 12 - x = 0

x = 12

So the width of the rectangle should be 12 feet, and its height would be 12 - (12/2) = 6 feet. This would maximize the amount of light entering the rectangular part of the window, given the 12 feet of frame material available.

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The solution set of the inequality 3x-2 < 2-2x is:

(4/5, ∞)

To find this we need to isolate the variable in the inequality.

Here we have:

3x - 2 < 2 - 2x

add 2x in both sides and add 2 in both sides, then we will get:

3x + 2x < 2 + 2

5x < 4

Now we can divide both sides by 5 to get:

x < 4/5

That is the inequality solved.

Then the solution set of the inequality is:

(4/5, ∞)

The set of all real numbers larger than 4/5.

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The area of the base of the prism is,

Area = 5.5 cm²

We have to given that,

This trapezoid-based right prism has a volume of 30 cm³.

We have;

Here we assume

a = 6

b = 5

c = 1

Now we know that

Area = (a + b) c / 2

Area = (6 + 5) 1 /2

Area = 11/2

Area = 5.5 cm²

Thus, The area of the base of the prism is,

Area = 5.5 cm²

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The researcher should analyze the data obtained from 933 people who were asked about their favorite type of TV program, with the condition that they could only choose one answer. The appropriate statistical test to analyze these data is c. Chi-square test.

The Chi-square test is used for analyzing categorical data, which is the case in this scenario where individuals have to choose among news, documentary, soap, or sports. The test will help the researcher determine if there is a significant difference in preferences for TV program types among the respondents.

The specific techniques and statistical tests used may vary depending on the goals of the research and the nature of the data.

Therefore option c is correct.

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c. rotate it 90° counterclockwise about point (-3,-3), then translate it 8 units to the right

The correct pair of transformations that moves quadrilateral 1 to quadrilateral 2 is option d: translate it 8 units to the right, then reflect it over the line y = -3.

First, by translating it 8 units to the right, all points of quadrilateral 1 will shift horizontally to the right by 8 units, maintaining their relative positions.

Next, reflecting it over the line y = -3 will result in a vertical flip of the shape. This reflection will change the orientation of the quadrilateral while keeping the translated positions intact.

Together, these two transformations will precisely move quadrilateral 1 to quadrilateral 2. Therefore, option d is the correct choice.

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The final amount Lochlon will earn on his investment after 5 years of compound interest is $1,104.36

Compound interest is a type of interest calculation where the interest earned is added to the principal amount, and the resulting sum becomes the new principal for the next interest calculation. The formula for compound interest is:

A = [tex]P(1 + r/n)^(^n^t^)[/tex]

Where:

A is the final amount including the interest

P is the principal amount

r is the annual interest rate as a decimal

n is the number of times the interest is compounded per year

t is the time in years

In this case, Lochlon transferred his investment into a money market account that earns compound interest of 1.95% annually, with a maturity date of 5 years.

To find the final amount Lochlon will earn, we need to know the principal amount, the interest rate, the number of times the interest is compounded per year, and the time period.

Assuming Lochlon invests a principal amount of P dollars, with an annual interest rate of r = 1.95%, and the interest is compounded annually (n = 1) for a time period of 5 years (t = 5), the formula for calculating the final amount (A) is:

A = [tex]P(1 + r/n)^(^n^t^)[/tex]

= [tex]P(1 + 0.0195/1)^(^1^*^5^)[/tex]

= [tex]P(1.0195)^5[/tex]

if Lochlon invests $1,000, for example, then his final amount (A) after 5 years would be:

A = [tex]1000(1.0195)^5[/tex]

= 1000(1.10436)

= $1,104.36

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The per unit total cost at an output level of 1,000 bicycles is calculated to be $139.13.

To calculate the per unit total cost using the high-low method, follow these steps:

1. Identify the highest and lowest output levels (1,150 and 950 bicycles).
2. Calculate the difference in variable costs and output levels: ($115,000 - $95,000) / (1,150 - 950) = $20,000 / 200 = $100 per bicycle.
3. Calculate the variable cost for 1,000 bicycles: $100 x 1,000 = $100,000.
4. Add the fixed cost: $100,000 (variable cost) + $45,000 (fixed cost) = $145,000 (total cost).
5. Calculate the per unit total cost: $145,000 / 1,000 = $139.13 per bicycle.

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5764801/256, is equivalent expression of [tex](7/2)^8[/tex] which is an exact value and cannot be simplified any further.

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable. You can write equivalent expressions by combining like terms. Like terms are terms that have the same variables raised to the same powers.

We can simplify the expression[tex](7/2)^8[/tex]by raising both the numerator and denominator to the 8th power:

We get,

[tex](7/2)^8 = 7^8 / 2^8[/tex]

To solve this value,  simplify the numerator and denominator separately:

So we get,

[tex]7^8[/tex]= 5764801

[tex]2^8[/tex]= 256

Therefore, [tex](7/2)^8[/tex] = 5764801/256, which is an exact value and cannot be simplified any further.

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Answer:

425

Step-by-step explanation:

212,5*2=425

The vertices of triangle XYZ are (-4, 0), (1, -2), and (-2, 1).

We are given that X''(4, 0), Y''(2, -1), and Z''(2, 1). We can use these coordinates to determine the coordinates of the vertices of triangle XYZ.

Starting with X, we have (-y, x) = (4, 0). This implies that y = 0 and x = -4.

Moving on to Y we have (-z, y) = (2, -1). This implies that z = -2 and y = 1.

Finally, for Z, we have (-x, z) = (2, 1). This implies that x = -2 and z = 1.

Therefore, the vertices of triangle XYZ are (-4, 0), (1, -2), and (-2, 1).

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The maximum vertical distance between the line y = x + 72 and the parabola y = x^2 is 518.67 units.

To find the maximum vertical distance between the line and the parabola, we need to find the point(s) where the distance is maximum.

The line y = x + 72 is a straight line with slope 1, and it intersects the y-axis at 72.

The parabola y = x^2 is a symmetric curve with vertex at (0,0).

To find the point(s) where the distance is maximum, we can find the intersection point(s) of the line and the parabola.

Substituting y = x + 72 in the equation of the parabola, we get x^2 - x - 5184 = 0.

Solving for x using the quadratic formula, we get x = (1 ± sqrt(1 + 20736))/2.

The two intersection points are (108, 180) and (-107, 65).

The maximum vertical distance between the line and the parabola is the difference between the y-coordinates of these points, which is approximately 518.67 units.

Therefore, the maximum vertical distance between the line y = x + 72 and the parabola y = x^2 is 518.67 units.

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To find dy/dt, we need to use implicit differentiation.

First, we differentiate both sides of the equation with respect to t:

2x(dx/dt) + 2y(dy/dt) = 2(dx/dt) + 4(dy/dt)

Next, we plug in the given values for x, y, and dx/dt:

2(3)(7) + 2(1)(dy/dt) = 2(7) + 4(dy/dt)

Simplifying, we get:

42 + 2(dy/dt) = 14 + 4(dy/dt)

Subtracting 2(dy/dt) and 14 from both sides:

28 = 2(dy/dt)

Finally, we divide both sides by 2 to solve for dy/dt:

dy/dt = 14
To find dy/dt, first differentiate the given equation x^2+y^2=2x+4y with respect to time t. Use the chain rule:

2x(dx/dt) + 2y(dy/dt) = 2(dx/dt) + 4(dy/dt).

Now substitute the given values, x = 3, y = 1, and dx/dt = 7:

2(3)(7) + 2(1)(dy/dt) = 2(7) + 4(dy/dt).

Solve for dy/dt:

42 + 2(dy/dt) = 14 + 4(dy/dt).

Rearrange and solve:

2(dy/dt) - 4(dy/dt) = 14 - 42,

-2(dy/dt) = -28.

Finally, divide by -2:

dy/dt = 14.

So the value of dy/dt is 14 when x = 3, y = 1, and dx/dt = 7.

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The measure of angle θ, to the nearest tenth of a degree is 233.1°.

To find the measure of angle θ, we need to use trigonometry. We can see that the point (-5,-6) lies in the third quadrant since both x and y coordinates are negative. We can draw a right-angled triangle with the origin (0,0) as the vertex and the given point (-5,-6) as one of the vertices on the x-y plane.

The hypotenuse of this triangle will be the distance between the origin and the point (-5,-6), which can be calculated using the Pythagorean theorem.

Using the Pythagorean theorem, we get:

√(5²+6²) = √(25+36) = √61

Now we can use trigonometry to find the measure of angle θ. We can see that the sine of θ is equal to the opposite side over the hypotenuse and the cosine of θ is equal to the adjacent side over the hypotenuse. So we have:

sin θ = -6/√61 and cos θ = -5/√61

Using a calculator, we can find that θ is approximately 233.1° to the nearest tenth of a degree.

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The probabilities are given as follows:

a) Square: 1/6.

b) Not the triangle: 43/48.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The area of the rectangle is given as follows:

A = 12 x 8 = 96.

(area of rectangle, multiply the dimensions).

The areas of each figure are given as follows:

Hence the probability of the square is given as follows:

p = 16/96 = 1/6.

(area of square divided by total area).

The probability that the region is not the triangle is given as follows:

p = (96 - 10)/96

p = 86/96

p = 43/48.

(triangle and not triangle are complementary events).

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Answer:   C

Step-by-step explanation:

f moved left 4 spaces in x direction to get to g

so take opposite sign

f(x+4)

To approximate √125.04 using linear approximation, first find the linearization of f(x) = ³√x at x = 125. Then use the point-slope form of the equation to find the equation of the tangent line and plug in x = 125.04 to get the approximation.

To approximate √125.04 using linear approximation and the function f(x) = ³√x, first find the linearization of f(x) at x = 125 in the form y = mx + b. Calculate f(125) and f'(x).Calculate f'(125): Use the point-slope form of the equation
1: Calculate f(125) and f'(x).
f(125) = ³√125 = 5
f'(x) = (1/3)x^(-2/3)
2: Calculate f'(125).
f'(125) = (1/3)(125)^(-2/3) = 1/15
3: Use the point-slope form of the equation y - y1 = m(x - x1) to find the equation of the tangent line.
y - 5 = (1/15)(x - 125)
4: Rearrange to find y in terms of x.
y = (1/15)(x - 125) + 5
5: Determine the values of m and b.
m = 1/15
b = (1/15)(-125) + 5
6: Plug in x = 125.04 to approximate √125.04.
²√125.04 ≈ (1/15)(125.04 - 125) + 5
The linearization of f(x) at x = 125 is y = (1/15)x + b, with m = 1/15 and b = (1/15)(-125) + 5. Using these values, the approximation of √125.04 is (1/15)(125.04 - 125) + 5.

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Answer: attach an image

Step-by-step explanation:

To find the principal, we can use the formula for simple interest:

I = P*r*t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

We need to convert 82 days to years by dividing it by 365 (the number of days in a year):

t = 82/365

t = 0.2247

Now we can plug in the values we know and solve for P:

67.14 = P*0.0625*0.2247

P = 67.14/(0.0625*0.2247)

P = 1900

Therefore, the principal is $1900.

Once we apply one of these tests and determine whether the series converges or diverges, we can then further analyze the series to find its sum, if it exists.

To test the convergence of a series, we typically use one of several tests, depending on the nature of the series. Some of the commonly used tests are:

Divergence test: If the terms of a series do not approach zero, the series must diverge. The test states that if lim(n->inf) an != 0, then the series diverges.

Comparison test: If the terms of a series are positive and can be compared with a known convergent or divergent series, we can determine the convergence or divergence of the given series. If an <= bn for all n and the series sum of bn converges, then the series sum of an also converges. If an >= bn for all n and the series sum of bn diverges, then the series sum of an also diverges.

Limit comparison test: If the terms of a series are positive, we can compare the given series with a known convergent series, using the limit comparison test. If lim(n->inf) (an/bn) = L, where L is a positive finite number, then both series either converge or diverge.

Ratio test: If the terms of a series approach zero and the ratio of consecutive terms approaches a limit L, then the series converges if L < 1 and diverges if L > 1. If L = 1, the test is inconclusive.

Root test: If the terms of a series approach zero and the nth root of the absolute value of the nth term approaches a limit L, then the series converges if L < 1 and diverges if L > 1. If L = 1, the test is inconclusive.

Alternating series test: If the terms of a series alternate in sign and decrease in magnitude, then the series converges.

There are also other tests, such as integral test, p-series test, and Dirichlet test, among others, which can be used to test the convergence of certain types of series.

Once we apply one of these tests and determine whether the series converges or diverges, we can then further analyze the series to find its sum, if it exists.

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The general solution to the differential equation is, |3x^4/(y(x+y))|x + |x^3| ln|y| + |x^3| ln|x+y| + h(x) = C.

The partial derivative of (3xy + y^2) with respect to y is 6xy + 2y, and the partial derivative of (x^2 + xy) with respect to x is 2x + y. Since these are not equal, the differential equation is not exact.

To make it exact, we need to find an integrating factor μ(x, y) such that μ(x, y)(3xy + y^2) dx + μ(x, y)(x^2 + xy) dy = 0 is exact. We can find μ(x, y) by using the formula:

μ(x, y) = e^(∫(∂M/∂y - ∂N/∂x)/N dx)

where M = 3xy + y^2 and N = x^2 + xy. We have:

(∂M/∂y - ∂N/∂x)/N = (6xy + 2y - 2x - y)/(x^2 + xy) = (6xy - x - y)/(x^2 + xy)

We can now find the integrating factor μ(x, y) by integrating this expression with respect to x:

μ(x, y) = e^(∫(6xy - x - y)/(x^2 + xy) dx) = e^(3ln|x| - ln|y| - ln|x+y| + C) = e^(ln|x^3/(y(x+y))| + C) = |x^3/(y(x+y))|e^C

where C is the constant of integration.

Now we multiply the original differential equation by the integrating factor μ(x, y) to obtain:

|3x^4/(y(x+y))| dx + |x^3/(y(x+y))| dy = 0

This is now an exact differential equation, and we can find its solution by integrating with respect to x or y. Integrating with respect to x, we get:

|3x^4/(y(x+y))|x + g(y) = C

where g(y) is the constant of integration. To find g(y), we integrate the coefficient of dy:

g(y) = ∫|x^3/(y(x+y))| dy = |x^3| ln|y| + |x^3| ln|x+y| + h(x)

where h(x) is another constant of integration. Substituting g(y) back into the solution, we have:

|3x^4/(y(x+y))|x + |x^3| ln|y| + |x^3| ln|x+y| + h(x) = C

This is the general solution to the differential equation.

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The equation 10S + 8W = 192 represents the calories Bridget ate on Monday and the equation 20S + 1W = 99 represents the calories she ate the next day.

The number of calories in each strawberry is 4, and the number of calories in each vanilla wafer cookie is 19.

a. We have two equations for the two days, using S for the number of calories in a strawberry and W for the number of calories in a vanilla wafer cookie:

On Monday:
10S + 8W = 192

On Tuesday:
20S + 1W = 99

b. To solve the system of equations using the substitution method, first solve one of the equations for one of the variables. We'll choose the second equation and solve for W:

W = 99 - 20S

Now substitute this expression for W in the first equation:

10S + 8(99 - 20S) = 192

Expand and simplify:

10S + 792 - 160S = 192

Combine like terms:

-150S = -600

Now divide by -150:

S = 4

Now that we have the value for S, substitute it back into the expression for W:

W = 99 - 20(4)

W = 99 - 80

W = 19

So the number of calories in each strawberry is 4, and the number of calories in each vanilla wafer cookie is 19. The solution is (S, W) = (4, 19).

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To evaluate the integral of tan inverse v(x+2) dx, we need to make a substitution. Let u = x + 2, then du/dx = 1 and dx= du. Therefore, the final answer is: ∫ tan inverse v(x+2) dx = (x+2) tan inverse v(x+2) - tan inverse v(x+2) / v'(x+2) + C

Substituting this back into the integral, we get:
∫ tan inverse v(x+2) dx = ∫ tan inverse v(u) du
Using the formula from the table of integrals, we have:
∫ tan inverse v(u) du = u tan inverse v(u) - ∫ u / (1 + v(u)^2) du
Substituting back u = x + 2, we get:
∫ tan inverse v(x+2) dx = (x+2) tan inverse v(x+2) - ∫ (x+2) / (1 + v(x+2)^2) dx
Now, we can use another substitution, let t = v(x+2), then dt/dx = v'(x+2) and dx = dt / v'(x+2).
Substituting this back into the integral, we get:
∫ (x+2) / (1 + v(x+2)^2) dx = ∫ (x+2) / (1 + t^2) dt / v'(x+2)
Using the formula from the table of integrals, we have:
∫ (x+2) / (1 + t^2) dt = tan inverse t + C
where C is the constant of integration.
Substituting back t = v(x+2), we get:
∫ (x+2) / (1 + v(x+2)^2) dx = tan inverse v(x+2) / v'(x+2) + C
Therefore, the final answer is:
∫ tan inverse v(x+2) dx = (x+2) tan inverse v(x+2) - tan inverse v(x+2) / v'(x+2) + C
where C is the constant of integration.

To evaluate the integral of tan inverse v(x+2) dx using substitution, we'll first make a substitution:
Let u = x+2. Then, du = dx.
Now, we can rewrite the integral as:
∫tan^(-1)(v(u)) du
Next, we'll look up the integral of tan^(-1)(v(u)) in a table of integrals. Unfortunately, there isn't a direct formula for this specific integral. However, we can use integration by parts to proceed further.
Let I = ∫tan^(-1)(v(u)) du. Let's choose:
f(u) = tan^(-1)(v(u)) and df(u) = du,
g'(u) = 1 and dg(u) = u du.
Using integration by parts formula:
I = f(u)g(u) - ∫g(u)df(u)
I = u*tan^(-1)(v(u)) - ∫u(1/(1+v^2(u))) du
Now, we'll need to substitute back x+2 for u:
I = (x+2)*tan^(-1)(v(x+2)) - ∫(x+2)(1/(1+v^2(x+2))) dx
This integral doesn't have a simple closed-form solution, so the final answer will remain in the form shown above.

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Answer:

[tex] \sqrt{ \frac{2}{3} } + \sqrt{6} \\ = \frac{ \sqrt{2} }{ \sqrt{3} } + \sqrt{6} \\ = \frac{ \sqrt{2} }{ \sqrt{3} } \times \frac{ \sqrt{3} }{ \sqrt{3} } + \sqrt{6} \\ = \frac{ \sqrt{6} }{3} + \sqrt{6} \\ = \frac{ \sqrt{6} }{3} + \frac{3 \sqrt{6} }{3} \\ = \frac{ 4\sqrt{6} }{3} [/tex]

The expression that describes the length of one side of the rooftop is therefore: x - 9.

In mathematics, an expression is a combination of one or more variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. An expression can be as simple as a single variable or constant, or it can be a more complex combination of variables and operations.

Here,

The expression for the area of the rooftop is (x + 9)², where x is a variable representing the length of one side of the rectangle. To find the factors of this expression, we can expand it using the identity (a+b)² = a² + 2ab + b².

Expanding (x + 9)², we get:

(x + 9)² = x² + 18x + 81

Now, we need to find the factors of this expression that are also factors of the length of the sides of the rectangle. Since the sides of the rectangle must have a common factor of x, we can factor out x from the expression:

x² + 18x + 81 = x(x + 18) + 81

The factors of (x + 9)² are x(x + 18) + 81, (x + 9)(x + 9), (x - 9)(x - 9), and -(x + 9)(x + 9).

Since we are looking for factors that represent the length of one side of the rooftop, we can eliminate the negative factor and the factor (x + 9)(x + 9), since the sides of a rectangle must be positive.

That leaves us with x(x + 18) + 81 and (x - 9)(x - 9).

The expression describes the length of one side of the rooftop: x - 9

This is because the sides of a rectangle must be positive, and (x - 9) is a factor of (x + 9)² that represents a positive length.

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