Answer to Question #298536 in Calculus for saNDY

Question

A firm faces the demand function

P = 190 - 0.6Q and the total cost function

C = 40 + 30Q + 0.4Q2

Answer 1

a)  What output will maximize profit?

$$MC=p$$ $$MC=C'(q)=30+0.8q$$

Then

$$30+0.8q=190-0.6q$$ $$q=114$$

b) What output will maximize total revenue?

$$TR=pq$$ $$TR=190q-0.6q^2$$ $$TR'(q)=190-1.2q$$ $$TR'(q)=0=>190-1.2q=0$$ $$q=158$$

c) What will the output be if the firm makes a profit of 4,760?

$$TR-TC=190q-0.6q^2-(40+30q+0.4q^2)$$ $$=160q-q^2-40$$

$$160q-q^2-40=4760$$ $$q^2-160q+4800=0, q\ge 0$$ $$(q-80)^2=1600$$ $$q_1=80+40=40, q_2=80-40=40$$