1. In this problem, we will analyze the profit found for sales of a certain product. A demand equation (sometimes called a demand curve) shows how much money people would pay for a product depending on how much of that product is available on the open market. Often, the demand equation is found empirically (through experiment, or market research).

a. Suppose that a market research company finds that at a price of p = $31, they would sell x = 74 boxes of product each month. If they lower the price to p = $25, then more people would purchase the product, and they can expect to sell x = 80 boxes in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. (Hint: Write an equation using two points in the form (x,p)).

A company’s revenue is the amount of money that comes in from sales, before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, x, by the demand equation, p.

b. Substitute the result you found from part a into the equation R = xp to find the revenue equation. Provide your answer in simplified form.

The costs of doing business for a company can be found by adding fixed costs, such as rent, insurance, and wages, and variable costs, which are the costs to purchase the product you are selling. The portion of the company’s fixed costs allotted to this product is $320, and the supplier’s cost for materials is $3 each. Let x represent the number of product boxes.

c. If b represents a fixed cost, what value would represent b?

d. Find the cost equation for the product. Write your answer in the form C = mx + b.

The profit made from the sale of the product is found by subtracting the costs from the revenue.

e. Find the Profit Equation by substituting your equations for R and C in the equation . Simplify the equation.

f. What is the profit made from selling 24 boxes of product per month?

g. What is the profit made from selling 29 boxes of product each month?

h. What is the profit made from selling no boxes of product each month? Interpret your answer.

i. Use trial and error to find the quantity of product boxes per month that yields the highest profit.

j. How much profit would you earn from the number you found in part i?

k. At what price would you sell the product to realize this profit (hint, use the demand equation from part a)?

a. To find the equation of the demand curve, we can use the two given points (74, 31) and (80, 25). The demand equation can be written in the form p = mx + b, where p represents the price and x represents the quantity of product sold.

Using the two points, we can find the slope (m) and the y-intercept (b) of the line:
m = (31 - 25) / (74 - 80) = -6 / -6 = 1
Using the point-slope form of a linear equation:
p - 31 = 1(x - 74)
p - 31 = x - 74
p = x - 74 + 31
p = x - 43

Therefore, the equation of the demand curve is p = x - 43.

b. The revenue equation is given by R = xp, where R represents the revenue and x represents the quantity of product sold. Substituting the demand equation p = x - 43 into the revenue equation:
R = x(x - 43)
R = x^2 - 43x

Therefore, the revenue equation is R = x^2 - 43x.

c. Given that b represents a fixed cost, the value of b would be the portion of the company's fixed costs allocated to this product, which is $320.

d. The cost equation for the product can be written in the form C = mx + b, where C represents the cost and x represents the quantity of product boxes. The variable costs for each product box are $3, and the fixed cost is $320. Therefore,
C = 3x + 320.

e. The profit equation is given by P = R - C. Substituting the revenue equation R = x^2 - 43x and the cost equation C = 3x + 320, we have:
P = (x^2 - 43x) - (3x + 320)
P = x^2 - 43x - 3x - 320
P = x^2 - 46x - 320

f. To find the profit from selling 24 boxes of product per month, we substitute x = 24 into the profit equation:
P = (24)^2 - 46(24) - 320
P = 576 - 1104 - 320
P = -848

The profit made from selling 24 boxes of product per month is -$848.

g. To find the profit from selling 29 boxes of product each month, we substitute x = 29 into the profit equation:
P = (29)^2 - 46(29) - 320
P = 841 - 1334 - 320
P = -813

The profit made from selling 29 boxes of product each month is -$813.

h. To find the profit from selling no boxes of product each month, we substitute x = 0 into the profit equation:
P = (0)^2 - 46(0) - 320
P = 0 - 0 - 320
P = -320

The profit made from selling no boxes of product each month is -$320. This means that the company would still incur the fixed costs of $320 even if they do not sell any product.

i. To find the quantity of product boxes per month that yields the highest profit, we can graph the profit equation and identify the maximum point. Alternatively, we can use calculus to find the maximum point by taking the derivative of the profit equation, setting it equal to zero, and solving for x.

j. Once we find the quantity of product boxes per month that yields the highest profit (let's say it is x = a), we substitute this value into the profit equation:
P = a^2 - 46a - 320

This will give us the maximum profit.

k. To find the price at which the company should sell the product to realize this maximum profit, we can use the demand equation p = x - 43. Substituting x = a (the quantity that yields maximum profit) into the demand equation will give us the price.