college algebra finding profit function

Question

For a certain company, the cost function for producing x items is C(x)=50x+150 and the revenue function for selling x items is R(x)=−0.5(x−120)2+7,200. The maximum capacity of the company is 140 items.

1Assuming that the company sells all that it produces, what is the profit function?

2The company can choose to produce either 70 or 80 items. What is their profit for each case, and which level of production should they choose?

Answer 1

R(x) = -0.5(x-120)^2 + 7,200= -0.5(x^2 -240x + 14400) + 7,200

= -.5x^2 + 120x + 7,200+7,200 = -x^2/2 + 120x + 14,400

C(x) = 50x + 150

P(x) = R(x)-C(x) = -x^2/2 + 70x + 14,250

maximum profit is when P'(x) = 0 = -x+70 = 0, x=70

they should produce 70 as it maximizes profit

P(70) = 14,250 + 4900/2 = 14,250+2,450 = $16,700

P(70)>P(80) = 14,250 - 6400/2 + 5600 = 14250-3200+5600= $16,650

profit is $50 more at 70 than at 80