A tent is in the shape of a cylinder surmounted by a cone. It is given that the radius of the base of the tent is 7 m, and the volume of the air inside the tent is 154 m³. Find the height of the cone (in metres), if it is half the value of the height of the cylinder.

Given:

Radius of the base of the tent = 7 m

Volume of the air inside the tent = 154 m3

Height of the cone = (1/2) × Height of the cylinder

Formulae Used:

Volume of a Right Circular cone = (1/3) × π × r² × h

Volume of a Right Circular cylinder = π × r2 × h

Calculation:

Let the required height of the cone be h metres

So, the height of the cylinder = 2h metres

The total volume of air inside the tent = Volume of the Cone + Volume of the Cylinder

⇒ 154 = [(1/3) × π × 72 × h] + [π × 72 × 2h]

⇒ 154 = π × 7² × h [(1/3) + 2] = 154

⇒ 154 = π × 72 × h [7/3] = 154

⇒ 154 = (22/7) × 7² × h [7/3]

⇒ 154 = 154 × h × [7/3]

⇒ h = 3/7

⇒ h = 0.428 ≈ 0.43 metre

∴ The required height of the cone is 0.43 metre