Question

Two cones have their heights in the ratio $1:3$ and radii in the ratio $3:1$. What is the ratio of their volumes?

Answer 1

Given,
Ratio of heights of two cones $=1:3$
Ratio of radii $=3:1$
Let $h$ and $3h$ be the height of two cones.
Also, $3r$ and $r$ be the corresponding radii of cones.
So, $r_1=3r,h_1=h,r_2=r,h_2=3h$.
Ratio of volumes $=\frac{\left(\frac{1}{3}\right)\pi r_1{}^2{h}_1}{\left(\frac{1}{3}\right)\pi r_2{}^2{h}_2}$
$\begin{array}{cc}& =\frac{(3r{)}^{2}h}{r^2\left(3h\right)}\\ & =\frac{9r^{2}h}{\left(3r^{2}h\right)}\\ & =\frac{3}{1}\end{array}$
Hence, the ratio of volumes $=3:1$