Question

A solid is hemispherical at the bottom and conical (of the same radius) above it. If the surface areas of the two parts are equal, then find the ratio of its radius and the slant height of the conical part.

Answer 1

$${\textbf{Step -1: Write the surface area of cone and of hemisphere}}{\textbf{.}}$$

$${\text{Surface area of cone = }}\pi {\text{rl}}$$

$${\text{Surface area of hemisphere = 2}}\pi {{\text{r}}^2}$$

$${\textbf{Step -2: Equalising them to find the ratio of radius and Slant Height}}{\textbf{.}}$$

$$\pi {\text{rl = 2}}\pi {{\text{r}}^2}$$

$$\Rightarrow {\text{l = 2r}}$$ $$\left[ {{\textbf{Cancelling }}\pi {\textbf{ from both side and one r from RHS}}} \right]$$

$$\therefore\dfrac{{\text{r}}}{{\text{l}}} = \dfrac{1}{2}$$

$${\textbf{ Hence, the ratio of }}\dfrac{{\textbf{r}}}{{\textbf{l}}}{\textbf{ is 1:2}}{\textbf{.}}$$