A television manufacturing company has two different factories. The profit function for Factory I when x televisions are manufactured and sold is as follows: $P ( x ) = - 0.07 x ^ { 2 } + 25 x - 250$ The profit function for Factory II when x televisions are manufactured and sold is as follows: $P ( x ) = - 0.075 x ^ { 2 } + 35 x - 1500$ For what number of televisions is the profit the same at both factories, and what is the profit?
A) 134 televisions; $1843
B) 233 televisions; $2583.33
C) 125 televisions; $1781.25
D) 0 televisions; $0

Question

Quizplus · Accepted Answer

The profit is the same when the profit functions are equal. So, we solve the equation:$$-0.07x^2+25x-250=-0.075x^2+35x-1500$$$$0.005x^2-10x+1250=0$$$$(x-250)(x-50)=0$$$$x=250,50$$Since the number of televisions must be positive, the profit is the same when 50 televisions are manufactured and sold.To find the profit, we evaluate either profit function at x = 50:$$P(50)=-0.07(50)^2+25(50)-250=-175+1250-250=825$$So, the profit is $825 when 50 televisions are manufactured and sold.

A Television Manufacturing Company Has Two Different Factories P(x)=−0.07x2+25x−250P ( x ) = - 0.07 x ^ { 2 } + 25 x - 250P(x)=−0.07x2+25x−250

A Television Manufacturing Company Has Two Different Factories $P ( x ) = - 0.07 x ^ { 2 } + 25 x - 250$