S=2700-1.25x Problem

Question

A company has determined that the demand function for a certain couch is given by s=2700−1.25x, where s is the selling price per couch, and x is the number of couches sold. The costs associated with producing a line of couches include a fixed cost of $100,000, plus each couch costs $220 to make. 

A) What is the revenue function R(x)? 

B) How many couches should be manufactured and sold in order to maximize revenue?

C) What is the cost function C(x) to produce x couches?

D) What is the profit function P(x)? 

E) How many couches should be manufactured and sold in order to maximize profit?

[NO CALCULUS]

Answer 1

a) R(x)=sx=2700x-1.25x²

b) dR(x)/dx=0 ⇒ 2700-2.5x=0 ⇒ 2.5x=2700 ⇒ x = 1080 ⇒ Max R(x) =1080

c) C(x) = fixed cost + per unit cost*quantity = 100 000 +220x

d) P(x)=R(x)-C(x)=2700x-1.25x²-(100 000 +220x)= -1.25x² +2480x -100 000

e) dP(x)/dx=0 ⇒ -2.5x+2480 = 0 ⇒ 2.5x = 2 480 ⇒ x = 992

WITH NO CALCULUS

b) for porabola R(x) = 2700x-1.25x² a vertex point's x = -b/2a ⇒ x= -2700/(2(-1.25))= 2700/2.5=1080

e) for porabola P(x) = -1.25x² +2480x -100 000 a vertex point's x=2480/2.5 = 992