Question

The total surface area of a cone whose radius is $\frac{r}{2}$ and slant height $2l$ is:

• A. $2\mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)$
• B. $\mathrm{\pi r}\left(\mathrm{l}+\frac{\mathrm{r}}{4}\right)$
• C. $\mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)$
• D. $2\mathrm{\pi r}$

Answer 1

The correct option is B

$\mathrm{\pi r}\left(\mathrm{l}+\frac{\mathrm{r}}{4}\right)$

Find the total surface area of the cone:

Given that, radius of the cone is $\frac{r}{2}$ and its height is $2l$

The total surface area of the cone of radius $R$ and slant height $L$ $=$ Curved surface area $+$ Base area

$$\begin{gathered} =\pi RL+\pi R^2 \\ =\mathrm{\pi }\left(RL+R^2\right) \\ =\mathrm{\pi }\left({\frac{\mathrm{r}}{2}\times 2\mathrm{l}+\left(\frac{\mathrm{r}}{2}\right)}^2\right) \\ =\mathrm{\pi }\left(\mathrm{rl}+\frac{{\mathrm{r}}^2}{4}\right) \\ =\mathrm{\pi r}\left(\mathrm{l}+\frac{\mathrm{r}}{4}\right) \end{gathered}$$

Hence, the total surface area of a cone whose radius is $\frac{r}{2}$ and slant height $2l$ is $\mathrm{\pi r}\left(\mathrm{l}+\frac{\mathrm{r}}{4}\right)$.

Therefore, the correct answer is option B.