Question

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

Answer 1

Let $r$ cm be the radius and $h$ cm the height of the cylindrical part. Then, $r=5cm,h=13cm$

Clearly radii of the spherical part and base of the conical part are also $r$ cm. Let $h_1$ cm be the height, $l$ cm be the slant height of the conical part.

Then
$l^2=r^2+h_1^2\Rightarrow l=\sqrt{r^2+h_1^2}\Rightarrow l=\sqrt{5^2+{12}^2}=13cm$

Now
surface area of the toy$=$
curved surface area of the cylindrical part + curved surface area of hemispherical part + curved surface area of conical part

$=(2\pi rh+2\pi r^2+\pi rl){cm}^2$

$=\pi r(2h+2r+l){cm}^2$

$=\{\frac{22}{7}\times 5\times (2\times 13+2\times 5+13)\}{cm}^2$

$=770{cm}^2$