14. A buoy is made in the form of a hemisphere surmounted by a right cone whose circular
base coincides with the plane surface of the hemisphere. The radius of the base of the
cone is 3.5 metres and its volume is2/3 of the hemisphere. Calculate the height of the
cone and the surface area of the buoy correct to 2 places of decimal.
Answers
Given:
A buoy is made in the form of a hemisphere surmounted by a right cone.
The radius of the base of the cone is 3.5 metres and its volume is 2/3 of the hemisphere.
To find:
Calculate the height of the cone and the surface area of the buoy correct to 2 places of decimal.
Solution:
From given, we have the data,
The radius of the base of the cone is 3.5 m = 7/2 m
Volume of a hemisphere = 2/3 × π × r³
= 2/3 × 22/7 × (7/2)³
= 539/6 m³
Volume of a right cone = 2/3 × Volume of a hemisphere
= 2/3 × 539/6
⇒ 1/3 × π × r² × h = 2/3 × 539/6
1/3 × 22/7 × (7/2)² × h = 2/3 × 539/6
∴ h = 14/3 = 4.67 m
∴ The height of the cone = 4.67 m
Surface area of buoy = 2πr² + πrl
l = √ (r² + h²) = √ [(7/2)² + (14/3)²] = 35/6 m
Therefore, the surface area of buoy = 2 × 22/7 × (7/2)² + 22/7 × 7/2 × 35/6
= 77 + 385/6 = 847/6
∴ The surface area of the buoy = 141.17 m²