Question

1. A firm has analyzed its operating conditions, prices, and cost have developed the following functions: R = 400 – 4q2 (per thousand naira) and Cost = q2 + 10q + 30 (per thousand naira) where q is the number of units produced and sold. If the firm wishes to maximize profit:

(a) What quantity should be sold and at what price?

(b) What will be the amount of profit?

2. Find the point of maximum value of the revenue function given as P = 400 – 4q. Hence, find the revenue.

Answer 1

1(a)

given

$p=400-4q^2\\TC=q^2+10q+30$

Marginal cost = change in total cost due to quantity

$=\frac{d(Total\space cost)}{d(quantity)}\\ = \frac{d( q^2 + 10q + 30)}{d( Q)}\\ = 2q+10$

Now, setting Price = Marginal cost,

We get, 

$400-4q^2 = 2q+10\\390=2q+4q^2$

$=> Q =9.6 units.$

$p=400-4(9.6)^2\\=31.36\space naira$

(b)

$TR=P\times Q\\=9.6\times31.36\\=301.06\\TC=(9.6)^2+10(9.6)+30\\=218.16\\profit=TR-TC\\=301.056-218.16\\=82.896$

2

$p=400-4q\\31.36=400-4q\\4q=400-31.36\\q=92.16\\TR=p\times Q\\=31.36\times92.16\\=2890.1376$