Question

A toy is in the form of a cone mounted on a hemisphere with the same radius. The diameter of the conical portion is $6$cm and its height is $4$cm. Determine the surface area and volume of the solid.

Answer 1

Conical portion
$r=3cm,h=4cm$
$l^2=r^2+h^2=3^2+4^2=9+16$
$l^2=25$
$l=\sqrt{25}=5$cm
Spherical portion $r=3cm$
Surface area of the toy$=$ CSA of hemisphere$+$ CSA of cone
$=2\pi r^2+\pi rl=2\times \frac{22}{7}\times 3\times 3c{m}^2+\frac{22}{7}\times 3\times 5c{m}^2$
$=\frac{396}{7}c{m}^2+\frac{330}{7}c{m}^2=\frac{726}{7}c{m}^2=103.71c{m}^2$
Surface area of the toy$=103.71c{m}^2$
Volume of the toy$=$ Volume of hemisphere$+$ Volume of cone
$=\frac{2}{3}\pi r^3+\frac{1}{3}\pi r^{2}h=\frac{2}{3}\times \frac{22}{7}\times 3\times 3\times 3c{m}^3+\frac{1}{3}\times \frac{22}{7}\times 3\times 3\times 4c{m}^3$
$=\frac{396}{7}c{m}^3+\frac{264}{7}c{m}^3=\frac{660}{7}c{m}^3$
$\therefore$ Volume of the toy$=94.28c{m}^3$.