Question

A cubical wooden block of side 7 cm is surmounted by a largest hemisphere. Find the volume and total surface area of the resulting solid.

Answer 1

Given:

Side of cube, a = 7 cm

Largest hemisphere on the cube ⇒ diameter = a = 7 cm

∴ r = `7/2` = 3.5 cm

1) Volume of the resulting solid

Volume = Volume of cube + Volume of hemisphere

(i) Volume of a cube

V1​ = a³ = 7³ = 343 cm³

(ii) Volume of a hemisphere

V₂​ = `2/3​πr^3 = 2/3​π (7/2)^3`

V₂ = `2/3 π xx 343/8 = (343π)/12`

Using π = `22/7`

`V_2 = 343/12 xx 22/7 = 539/6`

= 89.83 cm³

∴ V = `343 + 539/6 = 2597/6`

= 432.83 cm³

2) Total Surface Area of the resulting solid

The top circular area of the cube is covered by a hemisphere, so:

TSA = SA of cube − area of base circle + CSA of hemisphere

= 6a² − πr² + 2πr² = 6a² + πr²

= 6(7²) + π(3.5)²

= 6(49) + π(12.25)

= 294 + 12.25π

Using π = `22/7`

TSA = `294 + 12.25 xx 22/7 = 294 + 38.5`

Total Surface Area = 332.5 cm²