Question

If $ R$ is the radius of the base of the hat, then the total outer surface area of the hat is

• A. $\mathrm{\pi r}\left(2\mathrm{h}+\mathrm{R}\right)$
• B. $2\mathrm{\pi r}(\mathrm{h}+\mathrm{r})$
• C. $2\mathrm{\pi rh}+{\mathrm{\pi R}}^2$
• D. None of these

Answer 1

The correct option is C

$2\mathrm{\pi rh}+{\mathrm{\pi R}}^2$

Explanation for the correct option:

Find the required surface area.

The outer surface area $A_1$ of the cylindrical part of the hat is ${\mathrm{\pi r}}^2+2\mathrm{\pi rh}$.

The outer surface area $A_2$ of the base of the hat is $\mathrm{\pi }\left({\mathrm{R}}^2-{\mathrm{r}}^2\right)$.

$\Rightarrow A_2={\mathrm{\pi R}}^2-{\mathrm{\pi r}}^2$

So, the total outer surface area $A$ of the hat is $A_1+A_2$.

$$\begin{gathered} \Rightarrow A={\mathrm{\pi r}}^2+2\mathrm{\pi rh}+{\mathrm{\pi R}}^2-{\mathrm{\pi r}}^2 \\ \Rightarrow A=2\mathrm{\pi rh}+{\mathrm{\pi R}}^2 \end{gathered}$$

Hence, option $C$ is the correct answer.