Question

From a solid cylinder of height 12 cm and radius 5 cm, a conical cavity of height 12 cm and of base radius 5 cm is hollowed out. Find the volume and total surface area of the remaining solid.

Answer 1

Given:

Radius of cylinder r = 5 cm

Height of cylinder h = 12 cm

A conical cavity of the same height and base radius is hollowed out.

Volume of the remaining solid

Volume = Volume of cylinder − Volume of cone

= `πr^2h − 1/3 πr^2h`

= `(1 − 1/3)πr^2h = 2/3π(5)^2(12)`

= 200 π cm³

Taking `π = 22/7`

`200 × 22/7`

​= `4400/7`

​= 62874 ​cm³

Total surface area of the remaining solid

Surfaces included:

Curved surface of cylinder

Bottom circular base of cylinder

Curved surface of the conical cavity

Slant height of cone:

`l = sqrt(r^2 + h^2)`

`l = sqrt(5^2 + 12^2)`

`l = sqrt169`

l = 13 cm

TSA = 2πrh + πr² + πrl

= 2π(5) × (12) + π(5²) + π(5) (13)

= (120 + 25 + 65) π = 210 πcm²

Taking `π = 22/7`

210 × `22/7`

= 660 cm²

Volume = 628 `4/7` ​cm³

Total Surface Area = 660 cm²