Question

If sum of radius and height of a cylinder is 6, then its maximum volume is

• A. $32\pi$
• B. $16\pi$
• C. $8\pi$
• D. None of these

Answer 1

The correct option is A $32\pi$

$Formulaused$
$volumeofcylinder=\pi r^{2}h$
$Let\text{height of the cylinder be h and radius be r}$
$\text{r+h=6}\left(given\right)$
$Volume\text{of cylinder =}\pi {\text{r}}^{2}h$
$V=\pi r^2(6-r)$
$=6\pi r^2-\pi r^3$
$\frac{dV}{dr}=12\pi r-3\pi r^2$
$\frac{dV}{dr}=0$
$12\pi r-3\pi r^2=0$
$\Rightarrow 12\pi r=3\pi r^2$
$r=4$
$Hence$
$4+h=6$
$\Rightarrow h=2$
$Volume=\pi {\left(4\right)}^2\left(2\right)$
$=32\pi$