Question

A metallic sphere of radius $10.5cm$ is melted and then recast into small cones each of radius $3.5cm$ and height $3cm$. The number of such cones is

• A. $63$
• B. $126$
• C. $21$
• D. $130$

Answer 1

Answer: (B)

$126$

Radius of sphere $\left(r\right)=10.5cm=\frac{21}{2}cm$
$\therefore$ Volume $=\frac{4}{3}\pi r^3=\frac{4}{3}\pi \times \frac{21}{2}\times \frac{21}{2}\times \frac{21}{2}$
Let $n$ cones are formed
Radius of cone $\left(R\right)=3.5cm=\frac{7}{2}cm$
Height of cone $\left(h\right)=3cm$
Volume of $1$ cone $=\frac{1}{3}\pi R^{2}h$
$=\frac{1}{3}\pi \times \frac{7}{2}\times \frac{7}{2}\times 3=\frac{49}{4}\pi$
Volume of $n$ cones $=\frac{49n\pi }{4}$
Vol. of cylinder $=$ Vol. of $n$ cones
$\frac{4\pi }{3}\times \frac{21}{2}\times \frac{21}{2}\times \frac{21}{2}=\frac{49n\pi }{4}$
$\therefore n=2\times 3\times 21$
$n=126$ cones.