Capital function from the investment rate

Statement

The net investment rate is $I(t)=40\,t^{3/5}$ and the capital stock at $t=0$ is $75$. Find the capital function $K(t)$.

Solution

$K(t)=\int 40\,t^{3/5}\,dt=40\cdot\frac{5}{8}\,t^{8/5}+c=25\,t^{8/5}+c.$ From $K(0)=75$ we get $c=75$. Hence $\boxed{K(t)=25\,t^{8/5}+75}.$