Question

A right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled in identical cones of height 12 cm and diameter 6 cm having a hemi-spherical shape on the top. Find the number of cones required.

Answer 1

Given:

For right circular cylinder

Diameter = 12 cm

$radius{r}_1=\frac{12}{2}=6cm$ and $height{h}_1=15cm$

Volume of Cylindrical ice-cream container= $\pi r_1^2{h}_1$ = $\frac{22}{7}\times 6\times 6\times 15$ $\to \pi r_1^2{h}_1$ = $\frac{11880}{7}c{m}^3$ $\leftarrow$ ( volume of the icecream holder)

For cone,

Diameter = 6 cm

$Radius\left(r_2\right)=\frac{6}{2}=3cm$ & height $h_2$ = 12 cm

Radius of hemisphere = radius of cone= 3 cm

Volume of cone full of ice-cream= volume of cone + volume of hemisphere

= $\frac{1}{3}\pi \times r_2^2{h}_2+\frac{2}{3}\pi r_2^3=\frac{1}{3}\pi r_2^2{h}_2+2r_2^3$

= $\frac{1}{3}\frac{22}{7}\times 3_2\times 12+2\times 3)$

= $\frac{1}{3}\times \frac{22}{7}\times (9\times 12+2\times 27$

= $\frac{22}{21}\times 108+54$

= $\frac{22}{21}\times 162$

= $\frac{22\times 54}{7}$

= $\frac{1188}{7}c{m}^3$


Let n be the number of cones full of ice cream.


Volume of Cylindrical ice-cream container =n × Volume of one cone full with ice cream

$\frac{11880}{7}=n\times \frac{1188}{7}$

11880 = n × 1188

n = $\frac{11880}{1188}=10$

n = 10

Hence, the required Number of cones = 10