Question

A solid is in the form of a right circular cone mounted on a hemisphere. The diameter of the base of the cone, which exactly coincides with hemisphere, is 7 cm and its height is 8 cm. The solid is placed in a cylindrical vessel of internal radius 7 cm and height 15 cm. How much water, in cm³, will be required to fill the vessel completely?

Answer 1

Given,

Diameter of cone = 7 cm

Radius of cone (r) = `7/2`​ = 3.5 cm

Height of cone (h) = 8 cm

Radius of hemisphere = radius of cone = r = 3.5 cm

Radius of cylindrical vessel (R) = 7 cm

Height of cylindrical vessel (H) = 15 cm.

Volume of solid = Volume of cone + Volume of hemisphere

= `1/3​πr^2h + 2/3 πr^3 = πr^2(1/3​h + 2/3​r)`

= `22/7 × (3.5)^2 × (1/3​ × 8 + 2/3​ × 3.5)`

= `22 × 0.5 × 3.5 × (8/3 + 7/3​)`

= `38.5 × 15/3​`

= 38.5 × 5

= 192.5 cm³.

By formula,

Volume of cylindrical vessel = πR²H

= `22/7×7^2×15`

= 22 × 7 × 15

= 2310 cm³.

Volume of water required to fill the vessel completely = Volume of cylindrical vessel − Volume of solid

= 2310 − 192.5

= 2117.5 cm³.

Hence, volume of water required to fill the vessel completely = 2117.5 cm³.