SOLUTION: 1. In this problem, we analyze the profit found for sales of graphic t-shirts. A demand equation (sometimes called a demand curve) shows how much money people would pay for a produ

Question 402565: 1. In this problem, we analyze the profit found for sales of graphic t-shirts. 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 a market research company finds that at a price of p = $25, they would sell x = 60 shirts each month. If they lower the price to p = $15, then more people would purchase the shirts, and they can expect to sell x = 75 shirts 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). Show all your work.
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. Show all your work.

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 $600, and the supplier�s cost for a shirt is $18 each. Let x represent the number of shirts.

c. If b represents a fixed cost, what value would represent b? Show all your work.
d. Find the cost equation for the shirt. Write your answer in the form C = mx + b. Show all your work.
The profit made from the sale of shirts 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. Show all your work.

f. What is the profit made from selling 33 shirts per month? Show all your work.
g. What is the profit made from selling 27 shirts each month? Show all your work.

h. What is the profit made from selling no shirts each month? Interpret your answer. Show all your work.

i. Use trial and error to find the quantity of shirts per month that yields the highest profit. Show all your work.
j. How much profit would you earn from the number you found in part i? Show all your work.
k. What price would you sell the shirts at to realize this profit? Hint: Use the demand equation from part a. Show all your work.

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
I don't think I or any tutor will answer all 11+ parts to this question...but I'll get you started and you can finish the rest.

To find the linear equation, we have two ordered pairs (25, 60) and (15, 75) (here, the price is the x-variable, however normal demand curves list the number sold as the x-variable). The slope m is equal to so m = -3/2. To find the y-intercept replace m, as well as an ordered pair to obtain

60 = (-3/2)(25) + b --> 60 = -75/2 + b --> b = 195/2 = 97.5

The demand equation is therefore given by . You should be able to figure out the rest of the problems. Btw, part i) does not require trial and error, since the revenue is given by px, we have a quadratic equation that we can optimize using the vertex of the quadratic, or by the derivative.
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