Question

If h, S and V be the height curved surface area and volume of a cone respectively, then 3πVh³ – S²h² + 9V² is equal to ________.

Answer 1

Height of the cone = h

Let r be the radius and l be the slant height of the cone.

∴ l² = r² + h² .....(1)

Volume of the cone = V = $\frac{1}{3}\mathrm{\pi }{r}^{2}h$

Curved surface area of the cone = S = $\mathrm{\pi }$rl

$\therefore 3\mathrm{\pi }V{h}^3\mathit{-}{S}^2{h}^2+9V^2$

$=3\mathrm{\pi }\times \frac{1}{3}\mathrm{\pi }{r}^{2}h\times h^3-{\mathrm{\pi }}^2{r}^2{l}^2\times h^2+9\times \frac{1}{9}{\mathrm{\pi }}^2{r}^4{h}^2$

$={\mathrm{\pi }}^2{r}^2{h}^4-{\mathrm{\pi }}^2{r}^2\left(r^2+h^2\right)\times h^2+{\mathrm{\pi }}^2{r}^4{h}^2$ [Using (1)]

$={\mathrm{\pi }}^2{r}^2{h}^4-{\mathrm{\pi }}^2{r}^2{h}^4-{\mathrm{\pi }}^2{r}^4{h}^2+{\mathrm{\pi }}^2{r}^4{h}^2$

$=0$

Thus, the value of $3\mathrm{\pi }V{h}^3\mathit{-}{S}^2{h}^2+9V^2$ is 0.

If h, S and V be the height, curved surface area and volume of a cone respectively, then 3πVh³ – S²h² + 9V² is equal to ___0___.