Question

If $h,c$ and $v$ respectively, are the height, the curve surface and volume of the cone, then
$3\pi v{h}^3-c^2{h}^2+8v^2=0$

• A. True
• B. False

Answer 1

The correct option is A True

Curved surface area of a cone, $c=\pi rl$

volume of a cone, $V=\frac{1}{3}\pi r^{2}h$

L.H.S.$=3\pi V{h}^3-c^2{h}^2+9V^2$

$=3\pi h^3\left(\frac{1}{3}\pi r^{2}h\right)-(\pi rl{)}^2{h}^2+9{\left(\frac{1}{3}\pi r^{2}h\right)}^2$

$={\pi }^2{r}^2{h}^4-{\pi }^2{r}^2{h}^2{l}^2+9\left(\frac{1}{9}{\pi }^2{r}^4{h}^2\right)$

$={\pi }^2{r}^2{h}^4-{\pi }^2{r}^2{h}^2(r^2+h^2)+{\pi }^2{r}^4{h}^2$(since, $l^2=r^2+h^2$)

$={\pi }^2{r}^2{h}^4-{\pi }^2{r}^4{h}^2-{\pi }^2{r}^2{h}^4+{\pi }^2{r}^4{h}^2=0=$R.H.S.