Question

If $h$, $S$, and $V$ denote respectively the height, curved surface area, and volume of a right circular cone, then $9\pi V{h}^3-S^2{h}^2-9V^2$ is equal to _____. (Given: $r=h$)

• A.
  ${\pi }^2{r}^4$
• B.
  ${\pi }^2{r}^6$
• C.
  $1$
• D.
  $0$

Answer 1

The correct option is D

$0$

$h$ = height , $S$ = curved surface area and $V$ = volume of the cone

Let $r$ be the radius of the cone. Then

$S=\pi rl,wherel=\sqrt{r^2+h^2}\therefore S=\pi r\sqrt{r^2+h^2}V=\frac{1}{3}\pi r^{2}hNow,9\pi V{h}^3-S^2{h}^2-9V^2=9\pi \left(\frac{1}{3}\pi r^{2}h\right)h^3-{\pi }^2{r}^2(r^2+h^2)h^2-9\left(\frac{1}{9}{\pi }^2{r}^4{h}^2\right)=3{\pi }^2{r}^2{h}^4-{\pi }^2{r}^4{h}^2-{\pi }^2{r}^2{h}^4-{\pi }^2{r}^4{h}^2=0$[By substituting $r=h$]