Answer:
[tex]h=r=(\frac{900}{\pi})^{\frac{1}{3}}[/tex]
Step-by-step explanation:
let r be the radius of the cylinder and h the height:
[tex]900=\pi r^2 h\\\\h=\frac{900}{\pi r^2}[/tex]
The surface are as a function of the radius and the first derivative is calculated as;
[tex]f(r)=S(r)=\pi r^2 +2\pi rh \\\\f(r)=\pi r^2+2\pi \frac{900}{\pi r^2}\\\\=\pi r^2+\frac{1800}{r}\\\\f\prime(r)=2\pi r-\frac{1800}{r^2}=0\\\\r=(\frac{900}{\pi})^{\frac{1}{3}}\\\\h=\frac{900}{\pi r^2}\\\\=\frac{900}{\pi \frac{900}{\pi}^{\frac{2}{3}}}\\\\h=(\frac{900}{\pi})^{\frac{1}{3}}\\\\h=r=(\frac{900}{\pi})^{\frac{1}{3}}[/tex]
Hence, the least surface area is achieved when the height and radius of the cylinder is equal, [tex]h=r=(\frac{900}{\pi})^{\frac{1}{3}} \ cm[/tex]
