Question

If the marginal cost of manufacturing a certain item is given by C' (x) = $\frac{dC}{dx}$= 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dC}{dx}=2+0.15x \\ \Rightarrow dC=\left(2+0.15x\right)dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ C=2x+\frac{0.15}{2}{x}^2+K.....\left(1\right) \\ \mathrm{At}C\left(0\right)=100,\mathrm{we}\mathrm{have} \\ 100=2\left(0\right)+\frac{0.15}{2}{\left(0\right)}^2+K \\ \Rightarrow K=100 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}T\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ C=2x+\frac{0.15}{2}{x}^2+100 \\ \Rightarrow C=0.075x^2+2x+100 \end{gathered}$$