Question

The internal and external diameters of a hollow hemispherical shell are $6$cm and $10$cm respectively. It is melted and recast into a solid cone of base diameter $14$cm. Find the height of the cone so formed.

Answer 1

Hollow hemisphere: External radius, $R=\frac{10}{2}=5cm,$
Internal radius, $r=\frac{6}{2}=3cm$
Cone:$r=\frac{14}{2}=7cm,h=?$
Volume of hollow hemisphere$=$ Volume of cone
$\frac{2}{3}\pi (R^3-r^3)=\frac{1}{3}\pi r^{2}h$
$\therefore h=\frac{\frac{2}{3}\pi (R^3-r^3)}{\frac{1}{3}\pi r^2}=\frac{2(R^3-r^3)}{r^2}=\frac{2(5^3-3^3)}{7^2}=\frac{2(125-27)}{49}=2\times \frac{98}{49}=4$cm
$\therefore$ Height of the cone of formed$=4$cm.