Evelyn has a coupon that will reduce her grocery bill by 8%. If c represents the cost of Evelyn's groceries, which expression represents Evelyn's grocery bill?
a) c-0. 08
b) c+0. 92
c) 0. 08c
d) 0. 92c​

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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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.

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 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:

425

Step-by-step explanation:

212,5*2=425

The commission of the salesperson is $750 if he received a commission of 25% of the difference between the auction price and the dealership price.

The salesperson's commission can be calculated by first finding the dealership price, which is 30% more than the auction price of $10,000.

30% of $10,000 = $3,000

Dealership price = $10,000 + $3,000 = $13,000

Next, we need to find the difference between the dealership price and the auction price

$13,000 - $10,000 = $3,000

The salesperson's commission is 25% of this difference

25% of $3,000 = $750

Therefore, the salesperson's commission is $750.

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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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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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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 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 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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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 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 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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equivalent integral with the order of integration reversed ST 2-3 F(x,y) dydc = o g(y) F(x,y) dedy+ So k(y) F(x,y) dardy Jh(v) a- he C- f(y) = g(y) = h(g) = k(y) = By reversing the order of integration, you've found an equivalent integral to the original one provided.

step-by-step explanation to achieve this, using the terms "integral," "reversed," and "equivalent" in the answer.

Step 1: Identify the original integral
The original integral is given as ∫∫ F(x, y) dy dx, where the integration limits are not explicitly provided. In this case, let's assume the limits of integration for y are from a(x) to b(x), and for x, they are from c to d.

Step 2: Sketch the region of integration
To reverse the order of integration, it's helpful to sketch the region of integration, which is the area in the xy-plane where the function F(x, y) is being integrated.

Step 3: Determine the new limits of integration
After sketching the region, determine the new limits of integration by considering the range of x for a given y value, and the range of y values. Let's assume the new limits for x are from g(y) to h(y), and for y, they are from e to f.

Step 4: Write the equivalent reversed integral
Now, you can write the equivalent integral with the order of integration reversed. In this case, it will be ∫∫ F(x, y) dx dy, with the new limits of integration. The complete reversed integral will look like:

∫(from e to f) [ ∫(from g(y) to h(y)) F(x, y) dx ] dy

By reversing the order of integration, you've found an equivalent integral to the original one provided.

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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 cost of purchasing 2 pounds of sliced ham and 3 pounds of sliced turkey from the meat counter is $30.95.

The table provided shows the purchases made by two customers at a meat counter. To determine how much you will pay for 2 pounds of sliced ham and 3 pounds of sliced turkey, you need to first look at the prices listed in the table. For sliced ham, the price per pound is $4.99, and for sliced turkey, the price per pound is $6.99.

To calculate the cost of 2 pounds of sliced ham, you can multiply the price per pound ($4.99) by the number of pounds (2), which gives you a total cost of $9.98. Similarly, to calculate the cost of 3 pounds of sliced turkey, you can multiply the price per pound ($6.99) by the number of pounds (3), which gives you a total cost of $20.97.

Therefore, the total cost for 2 pounds of sliced ham and 3 pounds of sliced turkey would be $9.98 + $20.97 = $30.95.

In conclusion, by using the prices listed in the table, it is possible to determine the cost of purchasing 2 pounds of sliced ham and 3 pounds of sliced turkey from the meat counter. It is important to remember to multiply the price per pound by the number of pounds needed for each item, and then add the costs together to get the total price.

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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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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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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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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 differential equation to describe the situation is:
dP/dt = 0.000508 * P * (1 - P/732)

We can write a logistic differential equation to describe the rate of change of students who heard the rumor. The equation is:

dP/dt = k * P * (1 - P/M)

where dP/dt is the rate of change in the number of students who heard the rumor, k is a constant, P is the number of students who have heard the rumor at a given time, and M is the maximum capacity of the school (732 students).

At 12 PM, P = 227 and dP/dt = 24 students per hour. We can plug these values into the equation:

24 = k * 227 * (1 - 227/732)

Now, solve for k:

k ≈ 0.000508

So, the differential equation to describe the situation is:

dP/dt = 0.000508 * P * (1 - P/732)

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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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The angular velocity of the elbow during the elbow flexion exercise was approximately 523.81 degrees/s

To calculate the angular velocity of the elbow during an elbow flexion exercise, we'll use the information given about the relative angle at different times. Here's the step-by-step explanation:
First, find the change in relative angle: Δθ = Final angle - Initial angle = 120 degrees - 10 degrees = 110 degrees.
Next, find the change in time: Δt = Final time - Initial time = 0.71s - 0.5s = 0.21s.
Now, calculate the angular velocity: ω = Δθ / Δt = 110 degrees / 0.21s ≈ 523.81 degrees/s.
So, the angular velocity of the elbow during the elbow flexion exercise was approximately 523.81 degrees/s.

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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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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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The effective price paid by the company after taking account of hedging would be $63.50, which is the August futures price on July 1. Calculate the profit or loss on the futures contracts and subtract that from the spot price on July 1, to determine the effective.

By entering into futures contracts on March 1, the company was able to lock in the price of $59 for the commodity, when the spot price was $60 and the futures price was $59, the difference between the futures price and the spot price on March 1 was $1 ($60 - $59), so the company had to pay an extra $1 per unit to hedge its purchase.

When the spot price increased to $64 on July 1, the company was still able to purchase the commodity at the lower hedged price of $59, plus the cost of the futures contract, which resulted in an effective price of $63.50. Overall, hedging helped the company mitigate the risk of price volatility and ensured a more predictable cost for the commodity purchase.

Effective price = Spot price - Profit from futures contracts

Effective price = $64 - $0.50(The difference between the futures price and the spot price on July 1 was $0.50 ($64 - $63.50))

Effective price = $63.50 per unit

Therefore, the effective price paid by the company after taking into account hedging was $63.50 per unit.

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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)

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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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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