Answer to Question #144980 in Math for Fridah

Question

Two Certified Public Accountants have recently started to give business advice to their clients. Acting as consultants, they estimated the demand curve of a clients firm to be: AR = 200 8Q Where AR is average revenue in millions of shillings and Q is the output in units. Investigations of the client firms cost profile shows that marginal cost (MC) is given by: MC = Q² 28Q + 211 (in millions of shillings). Further investigations have shown that the firms cost when not producing output is Sh.10 million.
Required:
The equation of total cost (5 marks)
The equation of total revenue (2 marks)
An expression for profit (2 marks)
The level of output that maximizes profit (5 marks)
The equation of marginal revenue (2 marks)

Answer 1

The equation of total cost is the integral of MC:

$TC = 1/3×Q^3 + 14Q^2 + 211Q + 10.$

The equation of total revenue is:

$TR = AR×Q = 200Q - 8Q^2.$

An expression for profit is $\pi = TR - TC = -1/3×Q^3 - 22Q^2 - 11Q - 10.$

The equation of marginal revenue is:

$MR = TR'(Q) = 200 - 16Q.$

The level of output that maximizes profit is at

MR = MC,

$Q^2 + 28Q + 211 = 200 - 16Q,$

$Q^2 + 44Q + 1 = 0,$ $Q1 = \frac{-44 + 43.95} {2} = -0.025,$

$Q2 = \frac{-44 - 43.95} {2} = -43.98.$