Answer to Question #209797 in Calculus for Tati

Question

Given the total cost, TC = 4 +Q, and the demand function is P = 10 – 2Q:

(a) Obtain the total revenue function.

(b) Find the profit function in terms of Q.

(c) Find the quantity when the profit is maximized. What is the maximum profit?

Answer 1

(a)

$$TR=pQ=(10-2Q)Q$$

$$TR=10Q-2Q^2$$

(b)

$$P=TR-TC=10Q-2Q^2-(4+Q)$$

$$P=-2Q^2+9Q-4$$

(c)

$$P'(Q)=-4Q+9$$

Critical number(s)

$$P'(Q)=0=>-4Q+9=0$$

$$Q=2.25$$

$$P(2)=-2(2)^2+9(2)-4=6$$

$$P(3)=-2(3)^2+9(3)-4=5$$

The Profit has the absolute maximum with value of $6$ when the quantity $Q=2.$