A firm can produce 100 units per week. If its total cost function is $C = 500 + 1400 x$ dollars, and its total revenue function is $R = 1500 x - x ^ { 2 }$ dollars, find the maximum profit.
A)$3886
B)$2000
C)$7007
D)$5503
E)$5836

Question

Quizplus · Accepted Answer

The profit function is $P(x) = R(x) - C(x) = (1500x - x^2) - (500 + 1400x)$. Simplifying gives $P(x) = -x^2 + 100x - 500$. The maximum profit occurs at the vertex of this parabola, which is at $x = -b/(2a) = -100/(2 \times -1) = 50$. Substituting $x = 50$ into the profit function gives $P(50) = -(50)^2 + 100(50) - 500 = 2000$.

A Firm Can Produce 100 Units Per Week C=500+1400xC = 500 + 1400 xC=500+1400x

A Firm Can Produce 100 Units Per Week $C = 500 + 1400 x$