Please help!!!!

Question

The price P (in dollars) and the quantity X sold of a certain product obey the demand equation

x = -8p+ 160 0 < p <= 20

A) express the revenue R as a function of P

B) what is the revenue of the price is $16?

C) what price P that maximizes revenue?

D) What is the maximum revenue?

E) at a price of $10, how many units are sold?

F) what price should the company charge to earn at least $792 in revenue?

Answer 1

R = px = p(-8p+160) = -8p^2 +160p

R(16) = -8(16)^2 +160(16) = -2048 + 2560 = 512

R'= -16p + 160 = 0

p= 160/16 = 20/2 = 10 = revenue maximizing price

R(10) = -8(10)^2 + 160(10) = -800 + 1600 = 800 = maximum revenue

x=-8p+160

x =-8(10)+160 = -80 + 160 = 80

792 = -8p^2 +160p

8p^2 -160p + 792 = 0

2p^2 - 40p + 198 = 0

p^2 -20p + 99 = 0

(p-9)(p-11) = 0

p = 9 or 11 as the prices that give at least 792 in revenue

Answer 2

Hello, Arin,

Interesting problem. I was in the middle of answering when Raymond B. posted a correct answer. He did a great job on the answers, so I can only add some explanation, and a graph.

The equation for the revenue, which would be (R = p*x), is R= -8x² + 160p, or R = -8p(x² - 20). It is a parabola and the graph of that equation illustrates the fraught relationship between price and revenue. I plotted it such that the x-axis is the price and y axis is the revenue. As one raises the price, the revenue skyrockets initially, suggesting a price tolerance. But it quickly reaches an asymptote around the $10 mark, and then drops dramatically. The graph includes points below 0 and above 20, which are not in the domain of your problem, so the negative sales aren't real (unless the company is buying them back).

Raymond found the optimum price by taking the first derivative of the equation, which finds the equation's maximum. He also found the two points at which the revenue equals $792 by setting the revenue to 792, rearranging and factoring the equation. I didn't have the patience, nor the cleverness, to factor, so thanks to him for that. I can see from the graph that these points would have to be close together, as they are.

Bob