Question

If a sphere is placed inside a right circular cylinder so as to touch the top, base and the lateral surface of the cylinder. If the radius of the sphere is r, the volume of the cylinder and the sphere are in the ratio _________.

Answer 1

Radius of the sphere = r

∴ Volume of the sphere, V₁ = $\frac{4}{3}\mathrm{\pi }{r}^3$ .....(1)

If a sphere is placed inside a right circular cylinder so as to touch the top, base and the lateral surface of the cylinder, then the height of the cylinder is equal to the diameter of the sphere and the diameter of the base of cylinder is equal to the diameter of the sphere.

Let R be the radius and H be the height of the cylinder.

Height of the cylinder = Diameter of the sphere

⇒ H = 2r .....(2)

Diameter of the base of cylinder = Diameter of the sphere

⇒ 2R = 2r

⇒ R = r .....(3)

∴ Volume of the cylinder, V₂ = $\mathrm{\pi }$R²H = $\mathrm{\pi }$ × r² × 2r = 2$\mathrm{\pi }$r³ .....(4) [Using (2) and (3)]

Now,

$\frac{V_2}{V_1}=\frac{2\mathrm{\pi }{r}^3}{{\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^3}=\frac{3}{2}$

⇒ V₂ : V₁ = 3 : 2

Thus, the ratio of volume of cylinder to volume of sphere is 3 : 2.

If a sphere is placed inside a right circular cylinder so as to touch the top, base and the lateral surface of the cylinder. If the radius of the sphere is r, the volume of the cylinder and the sphere are in the ratio ___3 : 2___.