Question

Find the optimum level of output and profit from the cost function

TC = 50 + 6Q²

 and price

P = 100 – 4Q

Also derive marginal cost and marginal revenue

Answer 1

$TC=50+6Q^2$

$P=100-4Q$

$MC=\frac {d(TC)}{dQ}=0+12Q=12Q$

At profit max point; $100-4Q=12Q$

$=100=12Q+4Q$

$=\frac{100}{16}=Q$

Optimal level of output ; $Q=6.25$

ii. Total Revenue =Price$\times$Quantity

At $Q=6.25$

$TR=[100(4\times6.25)]\times6.25$

$=468.75$

Total cost (TC) $=50+6(6.25)^2$

$=284.375$

Therefore; $profit=TR-TC$

$=468.75-284.375$

$Profit=184.375$

Deriving Marginal cost and Marginal Revenue.

Given that; $P=100-4Q$

Multiply both sides by $Q$

$PQ=TR=100Q-4Q^2$

$=MR=\frac{d(TR)}{dQ}=(100\times1)-(4\times2)Q^2-^1$

$MR=100-8Q$

$MC=$ $\frac {d(TC)}{dQ}=0+12Q=12Q$

$MC=12Q$ .