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. 0.0
• B. 1
• C. 2
• D. infinite

Answer 1

The correct option is B 1

Let radius be '$r$' and height be '$h$' of the solid cone, where $r$ and $h$ are integers
Volume = surface area
$\frac{1}{3}\pi r^{2}h=\pi rl+\pi r^2$ [$l=\sqrt{r^2+h^2}$]
$\Rightarrow rh-3r=3\sqrt{r^2+h^2}$
Squaring both sides,
$\Rightarrow r^2{h}^2+9r^2-6r^{2}h=9(r^2+h^2)\Rightarrow r^2{h}^2-9h^2=6r^{2}h\Rightarrow h^2(r^2-9)=6r^{2}h\Rightarrow h=\frac{6r^2}{r^2-9}=6+\frac{54}{r^2-9}\Rightarrow r^2=9+\frac{54}{h-6}$
since $r$ and $h$ are integer, so $r^2$ will be perfect square,
for $r^2=16,h\ne I{r}^2=25,h\ne I{r}^2=36,h=8r^2=49,h\ne I$
Hence there is only 1 possible integer values of $r$ and $h$.