Question

Find the cost function if the marginal cost function is given by C'(x)=x^2/5+8

and 32 units costs $381

Answer 1

Answer:

The marginal cost function is given by:

$C'(x) = \frac{x^2}{5} +8$               ......[1]

Now, we can find the cost function by taking the antiderivative :

$C(x) = \int C'(x)$                ......[2]

Substitute [1] in [2] we get;

$C(x) = \int \frac{x^2}{5} +8$

Integration formulas:

• $\int x^n = \frac{x^{n+1}}{n+1}$

• $\int c = cx$    ; where c is constant.

then;

$C(x) = \frac{x^3}{5 \cdot 3} +8x + c_1$  where $c_1$ is constant

Simplify:

$C(x) = \frac{x^3}{15} +8x + c_1$              ......[3]

It is also given that for 32 units costs is $ 381

substitute the value of x = 32 and C(x) = $ 381, in [3] to find the constant term $c_1$

$C(32) = \frac{(32)^3}{15} +8(32)+ c_1$ or

$381 = \frac{(32)^3}{15} +8(32)+ c_1$

or

$381 = 2184.53333 + 256 +c_1$

$381 = 2440.53333+c_1$

Simplify:

$c_1 \approx -2059.53$

Substitute this value in [3] we get;

$C(x) = \frac{x^3}{15} +8x - 2059.53$

Therefore, the cost function is, $C(x) = \frac{x^3}{15} +8x - 2059.53$