Question

The total cost $C\left(x\right)$ in Rupees, associated with the production of $x$ units of an item is given by
$C\left(x\right)=0.005x^3-0.02x^2+30x+5000$
Find the marginal cost when $3$ units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Answer 1

As given, marginal cost is the instantaneous rate of change of total cost at any level of output.
Let $M.C$ be marginal cost, so

$M.C=\frac{dC}{dx}$

It is given $C\left(x\right)=0.005x^3-0.02x^2+30x+5000$

$M.C=\frac{d}{dx}(0.005x^3-0.02x^2+30x+5000)$

$\Rightarrow M.C=0.005\left(3x^2\right)-0.02\left(2x\right)+30$

When $3$ units are produced, putting $x=3$,

we get

$\Rightarrow M.C=0.005\left(27\right)-0.02\left(6\right)+30$

$\Rightarrow M.C=0.015+30=30.015$

Hence, the required marginal cost is $Rs.30.02$ (nearly).