Question

Base area of a right cone is $154{cm}^2$. If its slant height is $25cm$. Find the height of the cone.

Answer 1

Given,
Slant height of the cone (l) $=25cm$
Base area $=154{cm}^2$
A cone has a circular base,
$\begin{array}{cc}\therefore & \pi r^2=154\\ \Rightarrow & \frac{22}{7}\times r^2=154\end{array}$
$\Rightarrow r^2=\frac{154\times 7}{22}$
$\begin{array}{c}\Rightarrow r^2=7\times 7\\ r=\sqrt{7\times 7}=7cm\end{array}$
Let the height of the cone be $h$,
$\therefore$ $l^2=r^2+h^2$
$\begin{array}{cc}\Rightarrow & (25{)}^2=(7{)}^2+h^2\\ \Rightarrow & h^2=(25{)}^2-(7{)}^2\end{array}$
$\begin{array}{c}=(25+7)(25-7)\\ =32\times 18\\ h=\sqrt{32\times 18}\\ =\sqrt{16\times 2\times 2\times 9}\\ =4\times 2\times 3\\ =24cm\end{array}$
Hence, the height of the cone $=24cm$.