Question

A company manufactures and sells x television sets per month. The monthly cost and​ price-demand equations are ​C(x)=75,000+60x and p(x)=200-x/30, 0 less than or equals 'x' less than or equals6000.

​(A) Find the maximum revenue.
​(B) Find the maximum​ profit, the production level that will realize the maximum​ profit, and the price the company should charge for each television set.
​(C) If the government decides to tax the company ​$5 for each set it​ produces, how many sets should the company manufacture each month to maximize its​ profit? What is the maximum​ profit? What should the company charge for each​ set?

Answer 1

Final Answers:

(A) Maximum revenue is $680,000 at x = 3000.

(B) Maximum profit of $490,000 at x = 2500. Price should be $125 per set.

(C) With the tax, produce 2000 sets for $100 profit per set at $150 price.

Step-by-step explanation:

For part A, revenue (R) is found by multiplying the quantity sold (x) by the price (p(x)). The price-demand equation p(x) = 200 - x/30 gives the price at a given quantity. Maximizing R, we find x that maximizes p(x) * x, resulting in $680,000 when x = 3000.

Part B requires finding the profit equation, which subtracts the cost function C(x) from the revenue equation. Maximize this profit function to find the production level for maximum profit. The optimal production level is x = 2500, yielding a maximum profit of $490,000. To find the price, substitute x = 2500 into the price-demand equation, p(x) = 200 - x/30, resulting in a price of $125 per set.

In part C, with a $5 tax per set, the cost equation becomes C(x) = 75,000 + 65x. Follow the same steps to maximize profit but with the adjusted cost function. The new optimal production level is x = 2000, yielding a maximum profit of $100 per set at a price of $150.