Question

If h, S and V denote respectively the height, curved surface area and volume of a right circular cone, then $3{\mathrm{\pi Vh}}^3-{\mathrm{S}}^2{\mathrm{h}}^2+9{\mathrm{V}}^2$ is equal to

(a) 8

(b) 0

(c) 4$\mathrm{\pi }$

(d) 32${\mathrm{\pi }}^2$

Answer 1

Here we are asked to find the value for a given specific equation which is in terms of V, h and S representing the volume, vertical height and the Curved Surface Area of a cone.

We know $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ and S = $\mathrm{\pi rl}$.
Also, $l=\sqrt{r^2+h^2}$
Now, the given equation is $3{\mathrm{\pi Vh}}^3-{\mathrm{S}}^2{\mathrm{h}}^2+9{\mathrm{V}}^2$
So,
$$\begin{gathered} 3{\mathrm{\pi Vh}}^3-{\mathrm{S}}^2{\mathrm{h}}^2+9{\mathrm{V}}^2 \\ =3\mathrm{\pi }\left(\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}\right){\mathrm{h}}^3-{\left(\mathrm{\pi rl}\right)}^2{\mathrm{h}}^2+9{\left(\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}\right)}^2 \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{l}}^2{\mathrm{h}}^2+9\left(\frac{1}{9}{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2\right) \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^2{\left(\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2}\right)}^2+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \end{gathered}$$
$$\begin{gathered} ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^2{\left(\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2}\right)}^2+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^2\left({\mathrm{r}}^2+{\mathrm{h}}^2\right)+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \\ ={\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4-{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2-{\mathrm{\pi }}^2{\mathrm{r}}^2{\mathrm{h}}^4+{\mathrm{\pi }}^2{\mathrm{r}}^4{\mathrm{h}}^2 \\ =0 \end{gathered}$$

Hence the correct choice is option (b).