Yahoo Answers 04-13 pt 1
- Author:
- Michael Brown
1. A certain item can be produced at a unit cost of C(X)=10x. The demand function D(X)=90-0.02x where D is the price and x is the number of units.
a. how many units must be produced to maximize profit?
b. what is the price that gives the maximum profit?
c. what is the maximum profit?
Income=x(90-0.02x)=90x-0.02x^2
Profit=Income-Cost=90x-0.02x^2-10x=80x-0.02x^2
Profit is maximized at the vertex of the parabola (where the derivative equals zero) or 1/2way between the zeros.
Solve the profit function for 0 (this one is easy to factor)
80x-0.02x^2=0
.02x(4000-x)=0
x=0 or 4000-x=0
x=0, 4000
P'=80-0.04x
-0.04x=-80
x=2000
2000 units
D(2000)=90-0.02(2000)=50
P(2000)=80000