Question

The total surface area of the cone whose radius is $\dfrac{r}{2}$ and slant height is $2l$ is
(a) $\pi 2r\left( l+r \right)$
(b) $\pi r\left( l+\dfrac{r}{4} \right)$
(c) $\pi r\left( l+r \right)$
(d) $2\pi rl$

Answer 1

Hint: Use the fact that the total surface area of a cone with radius ‘r’ and slant height ‘l’ is $\pi r\left( r+l \right)$. Substitute the given values of radius and slant height in the formula and simplify the expression to calculate the total surface area of the given cone.

Complete step-by-step solution -
We have to calculate the total surface area of the cone whose radius is $\dfrac{r}{2}$ and slant height is $2l$.
We know that the total surface area of a cone with radius ‘r’ and slant height ‘l’ is $\pi r\left( r+l \right)$.

 

We know that the radius of the given cone is $\dfrac{r}{2}$ and its slant height is $2l$.
Substituting these values in the formula of calculating the total surface area of the cone, the total surface area of the given cone is $=\pi \dfrac{r}{2}\left( \dfrac{r}{2}+2l \right)$.
Simplifying the above expression, the total surface area of the cone is $=\pi \dfrac{r}{2}\left( \dfrac{r}{2}+2l \right)=\pi r\left( \dfrac{r}{4}+\dfrac{2l}{2} \right)=\pi r\left( \dfrac{r}{4}+l \right)$.
Hence, the total surface area of the cone whose radius is $\dfrac{r}{2}$ and slant height is $2l$ is $\pi r\left( \dfrac{r}{4}+l \right)$, which is option (b).

Note: One must be careful above units while calculating the volume of the cone. The units of volume of the cone is the cube of units of radius of the cone. The total surface area of the cone is the sum of the curved surface area and area of the base of the cone.