Question

A solid metallic sphere of diameter $21cm$ is melted and recast into a number of smaller cones, of diameter $3.5cm$ and height $3cm$. The number of cones so formed is:

• A. 405
• B. 540
• C. 504
• D. 254

Answer 1

The correct option is C 504

Given: Radius of the sphere$=\frac{21}{2}cm$
Volume of the sphere$=\frac{4}{3}\pi (\frac{21}{2}{)}^{3}c{m}^3$
Radius of the cone$=\frac{3.5}{2}=\frac{7}{4}cm$ and height$=3cm$
Volume of cone$=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi (\frac{7}{4}{)}^2\times 3c{m}^3$
Let the number of cones formed be $n$.
Then, $n\times \frac{1}{3}\pi \times (\frac{7}{4}{)}^2\times 3=\frac{4}{3}\pi \times (\frac{21}{2}{)}^3$
$\Rightarrow n\times (\frac{7}{4}{)}^2\times 3=4\times (\frac{21}{2}{)}^3$
$\Rightarrow n=\frac{4}{3}\times \frac{21}{2}\times (\frac{21\times 4}{2\times 7}{)}^2$
$\Rightarrow n=\frac{4}{3}\times \frac{21}{2}\times 6^2$
$\Rightarrow n=14\times 36$
$\therefore n=504$.