The ratio of the radius and height of a cone is 3 : 4. Its volume is…

The correct answer is Option (3) → 5 cm

The volume of a cone is \(\frac{1}{3}\pi r^2h\), and the radius, the height and the slant height \(l\) form a right triangle, so \(l=\sqrt{r^2+h^2}\). The volume fixes the actual size of the cone, whose shape the ratio 3 : 4 gives.

Assuming the radius to be \(3k\) and the height \(4k\):

• Volume \(\displaystyle =\frac{1}{3}\times\frac{22}{7}\times(3k)^2\times4k=\frac{264}{7}k^3\)

• \(\displaystyle 37\frac{5}{7}=\frac{264}{7}\), so \(k^3=1\) and \(k=1\)

• Radius \(=3\) cm, height \(=4\) cm

• Slant height \(=\sqrt{3^2+4^2}=\sqrt{25}=5\) cm