Question

Radius and height of a right circular cone and that of a right circular cylinder are respectively equal. If the volume of a cylinder is $120c{{m}^{3}}$, then the volume of the cone is equal to:
A) $120c{{m}^{3}}$
B) $360c{{m}^{3}}$
C) $40c{{m}^{3}}$
D) $90c{{m}^{3}}$

Answer 1

Hint: The radius of a right circular cone and a right circular cylinder is r cm. The height of the right circular cone and the right circular cylinder is h cm. Then volume of a right circular cylinder is $\pi {{r}^{2}}h$ cubic unit and the volume of a right circular cone is $\dfrac{1}{3}\pi {{r}^{2}}h$ cubic unit. Volume of the cylinder is given to us. Form an equation and then find out the volume of the cone.

Complete step-by-step answer:

It is given in the question that the radius and height of a right circular cone and that of a right circular cylinder are respectively equal.
Therefore, the radius of a right circular cone and a right circular cylinder are equal. Let the radius be r cm.
The height of a right circular cone and a right circular cylinder are also equal. Let the height be h cm.
Now the volume of the cylinder is given to us.
We know that if r is the radius of a cylinder and h is the height then the volume is:
$\pi {{r}^{2}}h$ cubic unit
Therefore,
$\pi {{r}^{2}}h=120.....(1)$
Now we need to find out the volume of the cone.
We know that if r is the radius of the base of a cone and h is the height then the volume is:
$\dfrac{1}{3}\pi {{r}^{2}}h$ cubic unit.
From equation (1) we have,
$\dfrac{1}{3}\pi {{r}^{2}}h=\dfrac{1}{3}\times 120=40$
Therefore the volume of the cone is $40c{{m}^{3}}$.
Hence, option (C) is correct.

Note: If the radius and the height are the same for a cone and a cylinder, the volume of a cylinder will always be greater than the volume of the cone. Therefore we can easily cancel out option (A) and option (b). The volume of a cone is one third of the volume of a cylinder if the base and height are the same. Therefore one third of 120 is 40. Hence, option (C) is correct.