Question

A container shaped like a right circular cylinder having diameter $12$ cm and height $15$ cm is full of ice cream. The ice cream is to be filled into cones of height $12$ cm and diameter $6$ cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

Answer 1

Given:
Height ($h_1$) of cylindrical container = 15 cm

Radius = $\frac{Diameter}{2}$

Radius ($r_1$) of circular end of container = $\frac{12}{2}=6cm$

Radius ($r_2$) of circular end of ice-cream cone = $\frac{6}{2}=3cm$
Height ($h_2$) of conical part of ice-cream cone = $12$ cm

Let $n$ ice-cream cones be filled with ice-cream of the container.

Volume of ice-cream in cylinder = n (Volume of 1 ice-cream cone + Volume of hemispherical shape on the top)

$\pi r_1^2{h}_1=n(\frac{1}{3}\pi r_2^2{h}_2+\frac{2}{3}\pi r_2^3)$

$\Rightarrow \pi \times 6^2\times 15=n(\frac{1}{3}\pi 3^2\times 12+\frac{2}{3}\pi 3^3)$

$\Rightarrow n=\frac{30\times 15}{\frac{1}{3}\times 9\times 12+\frac{2}{3}\times 27}$

$\Rightarrow n=\frac{36\times 15\times 3}{108+54}$

$n=10$

So, $10$ ice-cream cones can be filled with the ice-cream in the container.