Question

A sphere fits exactly inside a right circular cone with base area $154{cm}^2$ as shown. What is the radius of the sphere if the height and slant height of the cone are integers?

• A. $21/4cm$
• B. $41/2cm$
• C. $6cm$
• D. $51/4cm$

Answer 1

The correct option is A $21/4cm$

$\begin{array}{c}Basearea=154c{m}^2\\ \pi r^2=154\\ \frac{22}{7}\times r^2=154\\ r^2=7\times 7\\ r=7\\ l\&hareinteger\\ S_n\alpha =\frac{n}{l}=\frac{R}{OA}\\ r\left(n-r\right)=PR\\ R=r{\left(\frac{l-r}{1+r}\right)}^{\frac{1}{2}}\\ =7{\left(\frac{25-7}{25+7}\right)}^{\frac{1}{2}}\\ =7\times {\left(\frac{18}{32}\right)}^{\frac{1}{2}}\\ =7\times \frac{3}{4}\end{array}$