Answer
$C(x)=10x+\displaystyle \frac{x^{3}}{300,000}+100,000$
Work Step by Step
The marginal cost at production level of x is
$C'(x)=10+\displaystyle \frac{x^{2}}{100,000}$
So, $C(x)$ is an antiderivative:
$C(x)=\displaystyle \int(10+\frac{x^{2}}{100,000})dx$
$=10x+\displaystyle \frac{1}{100,000}(\frac{x^{3}}{3})+D$
The indefinite integral gives us a collection of functions.
To find the exact function, we find $D.$
Given the fixed cost (the cost regardless of production)
$C(0)=100,000$, we find $D.$
$100,000=0+0+D$
$D=100,000$
Thus,
$C(x)=10x+\displaystyle \frac{x^{3}}{300,000}+100,000.$
