Question

The number of solid cones with integer radius and integer height each having its volume numerically equal to its total surface area is

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (B)

Let height and radius of cone is $h$ and $r$ respectively, $h,r\in I$
Given volume of cone $=$ Surface area of cone
$\frac{1}{3}\pi r^{2}h=\pi rl+\pi r^2$
$\Rightarrow \frac{1}{3}\pi r^{2}h=\pi r\sqrt{h^2+r^2}+\pi r^2$
$\Rightarrow \frac{1}{3}rh=\sqrt{h^2+r^2}+r\left[r\ne 0\right]$
$\Rightarrow rh-3r=3\sqrt{h^2+r^2}$
$\Rightarrow r^2{h}^2+9r^2-6h{r}^2=9h^2+9r^2$
$\Rightarrow h^2\left(r^2-9\right)=6h{r}^2$
$\Rightarrow h=\frac{6r^2}{r^2-9}$
$\Rightarrow h=6\left(\frac{r^2}{r^2-9}\right)$
$\Rightarrow h=6+\frac{54}{r^2-9}$
$h$ and $r$ are integer.
$\therefore r^2-9$ is a factor of $54$.
$\therefore r^2-9=1,2,3,6,9,18,27,54$
$r^2=10,11,12,15,18,27,36,63$
$\therefore r=6$ only possible value.
$\therefore h=6+\frac{54}{36-9}$
$=6+2=8$
$\therefore r=6,h=8$