Functional Modeling Math Example 4

Solution

1. 1
   Volume constraint: $\pi r^2 h=500 \Rightarrow h=\frac{500}{\pi r^2}$. Surface area: $S=2\pi r^2+2\pi r h=2\pi r^2+2\pi r\cdot\frac{500}{\pi r^2}=2\pi r^2+\frac{1000}{r}$.
2. 2
   Minimize: $S'(r)=4\pi r - \frac{1000}{r^2}=0 \Rightarrow 4\pi r^3=1000 \Rightarrow r^3=\frac{250}{\pi} \Rightarrow r=\left(\frac{250}{\pi}\right)^{1/3}\approx4.30$ cm.