Question

What is the least number of solid metallic spheres, each 6 cm in diameter, that should be melted to cast a solid metallic cylinder whose height is 53 cm and diameter 6 cm? Also find the total surface area of the cylinder.

Answer 1

Given:

each sphere has diameter 6 cm ⇒ radius r = 3 cm. The cylinder has diameter 6 cm ⇒ radius R = 3 cm and height h = 53 cm

Volume of the cylinder

Vc = π R² h

= π × 32 × 53

= 477π cm³

Volume of one sphere Vs

= `(4/3)π r^3`

= `(4/3)π × 3^3`

= `36π cm^3`

Number of spheres required

n = `(Vc)/(Vs)`

= `(477π)/(36π)`

= `477/36`

= `53/4`

= 13.25

Since we cannot have a fractional sphere, the least whole number = 14 spheres.

Total surface area (TSA) of the cylinder

TSA = 2πR(R + h)

= `2π × 3 × (3 + 53)`

= 336π cm²

Using π = `22/7`

TSA = 336 × `22/7`

= 1056 cm2

Least number of spheres = 14

Total surface area of the cylinder = 1056 cm²