Question

$504$ cones, each of diameter $3.5\mathrm{cm}$ and height $3\mathrm{cm}$, are melted and recast into a metallic sphere. Find the diameter of the sphere.

Answer 1

Step1 : Find Volume of $504$ cones

Given,

Diameter of each cone $=3.5\mathrm{cm}$

Radius, $\mathrm{r}=\frac{3.5}{2}$

Height of each cone $=3\mathrm{cm}$

Thus the volume of each cone, $\mathrm{V}=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

$=\frac{1}{3}\pi {\left(\frac{3.5}{2}\right)}^2\times 2.3$

$=\frac{1}{3}\times \frac{22}{7}\times \frac{3.5}{2}\times \frac{3.5}{2}\times 3$

$=\frac{19.25}{2}{\mathrm{cm}}^3$

Thus, volume of $504$ cones $=504\times \frac{19.25}{2}=4851{\mathrm{cm}}^3$

Step 2: Find the diameter of the sphere

As per the given condition, $504$ cones are melted to form a sphere. Thus, the volume of $504$ cones and sphere made from them will be the same.

Hence,

Volume of sphere $=4851{\mathrm{cm}}^3$

$\frac{4}{3}\times \pi \times {\mathrm{r}}^3=4851$

${\mathrm{r}}^3=\frac{4851\times 3}{4\times \mathrm{\pi }}$

$\mathrm{r}=10.5\mathrm{cm}$

Therefore, the diameter of the sphere is, $\mathrm{d}=2\mathrm{x}10.5=21\mathrm{cm}.$