Question

A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on other. The radius and height of the cylindrical part are$5cm$ and$13cm$ respectively. The radii of hemisphere and conical parts are the same as that of the cylindrical part. Find the surface area of the toy if the total height of the conical part is$12cm$.

Answer 1

Hint: Calculate the surface area of all three shapes separately and later add. Use respective known formulas of surface area.
Now it is given that
Height of cylindrical part,$h$ $ = 13cm$
Radius of cylindrical part,$r$ $ = 5cm$
The radii of the spherical part and base of the conical part are also $r$.Let us suppose ${h_1}$ be the height of the conical part and $l$ be the slant height of the conical part.
We Know that,
$
  {l^2} = {r^2} + {h_1}^2 \\
   \Rightarrow l = \sqrt {{r^2} + {h_1}^2} \\
   \Rightarrow l = \sqrt {{5^2} + {{12}^2}} = 13cm \\
 $
Now, the surface area of the toy$ = $ curved surface area of the cylindrical part$ + $ curved surface area of the hemispherical part$ + $ curved surface area of the conical part.
$
   = \left( {2\pi rh + 2\pi {r^2} + \pi rl} \right)c{m^2} \\
   = \pi r\left( {2h + 2r + l} \right)c{m^2} \\
   = \left( {\dfrac{{22}}{7} \times 5 \times \left( {2 \times 13 + 2 \times 5 + 13} \right)} \right)c{m^2} \\
   = 770c{m^2} \\
 $
 Therefore, the surface area of the toy$ = 770c{m^2}$
Note- Whenever we face such types of questions the key concept is that we should write what is given to us and then use the formula that is suitable according to the question, like we did. Here, we find the slant height, the slant height of the cone and then knowing the values of curved surface area of the cylindrical part, curved surface area of the hemispherical part and curved surface area of the conical part we get our answer.