Answer to Question #210214 in Calculus for tati

Question

A market research company has derived the profit function for a Chackzy Inc as TP = -Q 2 + 60Q + 100

However, Mr John, the chairman of Chackzy Inc. needs more detailed information. You are assigned by the company to answer his questions which are as follows:

(a) At what level of output will my company maximize the profit?

(b) What will my company’s maximum profit be?

(c) What quantities will give my firm a breakeven state?

(d) Sketch the ‘TP’ graph, with clear indication of the maximum point as well as the breakeven points. Label the graph accordingly

Answer 1

The Total profit function is-

$TP=-Q^2+60Q+100$

(a) Differentiate TP w.r.t Q-

$\dfrac{d TP}{dQ}=-2Q+60$

For maximum profit-

$-2Q+60=0\implies Q=30$

Hence Profit is maximum at 30 level of output.

(b) The value of maximum profit is-

$TP_{max}=-(30)^2+60(30)+100=-900+1800+100=1000$

Hence Maximum profit is 1000.

(c) for breakeven value The value of Total profit should be 0

-$Q^2+60Q+100=0$

On solving the above equation we get-

Q=61.62=62

At 62 units we get the breakeven value.

(d) The Total profit graph is-