Question

A toy is in the form of a cone mounted on a hemisphere of radius $3.5cm$. If total height of the toy is $15.5cm$, find its total surface area.

Answer 1

Step 1: Find the slant height of the cone.

Let $h$ be the height of the cone and $r$ be the common radius of both hemisphere and cone.

We have, $r=3.5cm$ and $h=\left(15.5-3.5\right)cm=12cm$

According to Pythagoras theorem, $l^2=r^2+h^2$

$\Rightarrow l^2={\left(3.5\right)}^2+{\left(12\right)}^2$

$\Rightarrow l=\sqrt{12.25+144}=12.5cm$

Step 2: Find the curved surface area of both cone and hemisphere.

Curved surface area (C.S.A.) of the cone having radius $\left(r\right)$ and height $\left(h\right)$ $=\mathrm{\pi rl}$

$=\frac{22}{7}\times 3.5\times 12.5$ ( $\mathrm{\pi }=\frac{22}{7}$ )

$=137.5c{m}^2$

Curved surface area (C.S.A.) of the hemisphere having radius $\left(r\right)$ and height $\left(h\right)$ $=2{\mathrm{\pi r}}^2$

$=2\times \frac{22}{7}\times {\left(3.5\right)}^2$

$=2\times \frac{22}{7}\times 12.25$

$=77c{m}^2$

Step 3: Find the total surface area of the toy.

Therefore, total surface area (T.S.A.) of toy $=77+137.5=214.5c{m}^2$

Hence, the total surface area is $214.5c{m}^2$.