A right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled in identical cones of height 12 cm and diameter 6 cm having a hemi-spherical shape on the top. Find the number of cones required.

Given,

Diameter of the cylinder = 12 cm

So, its radius (r) = $\dfrac{12}{2}$ = 6 cm

Height of the cylinder (h) = 15 cm

Diameter of the cone = 6 cm

So, its radius (R) = $\dfrac{6}{2}$ = 3 cm

Height of the cone (H) = 12 cm

Radius of the hemisphere = Radius of cone = R = 3 cm

Now,

Let the number of cones be n.

Volume of the cylinder = πr²h

= π × (6)² × 15

= 540π cm².

Volume of an ice-cream cone with ice-cream = Volume of cone + Volume of hemisphere

$= \dfrac{1}{3}πR^2H + \dfrac{2}{3} \times π \times R^3 \\[1em] = \dfrac{1}{3} \times π \times 3^2 \times 12 + \dfrac{2}{3} \times π \times 3^3 \\[1em] = 36 π + 18 π \\[1em] = 54 π \text{ cm}^3.$

Let no. of cones required be n.

Volume of cylinder = n × Volume of each cone

n = $\dfrac{\text{Vol. of cylinder}}{\text{Vol. of ice-cream cone}} = \dfrac{540π}{54π}$ = 10.

Hence, the number of cones required = 10.