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. 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)?