Question

A solid metallic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter  4 \[\frac{2}{3}\] cm and height 3 cm. Find the number of cones so formed.

Answer 1

The radius of solid metallic sphere, `R= 28/2`= 14 cm

The volume of sphere

`= 4/3 pi R^3`

`= 4/3 xx pi xx (14)^3`

`= 4/3pi xx 14 xx 14 xx 14`

`= (10976 pi)/3 cm^3`

Given, the sphere is recast into smaller cones.

The radius of cone,

 `r = 14/(3 xx 2)`

`= 7/3 cm`

The height of cone h = 3 cm

Let n be the no. of smaller cones.

Clearly, the volume of solid sphere = n × volume of one smaller cone

\[\frac{10976}{3}\pi = n \times \frac{1}{3}\pi \times \left( \frac{7}{3} \right)^2 \times 3\]

\[n \times \frac{49}{3} = 10976\]

\[n = \frac{10976 \times 3}{49}\]

\[n = 672\]

Thus, the no. of smaller cones = 672