Question

A cylindrical container having diameter 16cm and height 40cm if full of ice-cream.
The ice-cream is to be filled into cones of height 12cm and diameter 4cm,
having a hemispherical shape on the top. Find the number of such cones which can
be filled with ice-cream.

Answer 1

Hint:- One complete ice-cream $ = $ Volume of cone $ + $ Volume of hemisphere. We need to remember the formula of volume of cylinder, cone and hemisphere.

Complete step-by-step solution -

As given in the question that,
$ \Rightarrow $Radius of the cylinder $ = {\text{ R }} = {\text{ }}8{\text{cm}}$
$ \Rightarrow $Height of the cylinder $ = {\text{ H }} = {\text{ }}40{\text{cm}}$
As, we know the formula to calculate volume of cylinder is $ = {\text{ }}\pi {R^2}H$
$ \Rightarrow $So, volume of given cylinder $ = {\text{ }}\pi {R^2}H{\text{ }} = {\text{ }}\pi *{(8)^2}*40$
$ \Rightarrow $Hence ice-cream that can be inside the cylinder $ = {\text{ }}\pi *{(8)^2}*40$ (1)
We are also given with the dimensions of the cone and that is,
$ \Rightarrow $Radius of cone $ = {\text{ }}r{\text{ }} = {\text{ }}2{\text{ }}cm$
$ \Rightarrow $Height of cone $ = {\text{ }}h{\text{ }} = {\text{ }}12{\text{ }}cm$
As, we know the formula to calculate volume of cone is $ = \dfrac{1}{3}\pi {r^2}h{\text{ }}$
$ \Rightarrow $Volume of cone $ = \dfrac{1}{3}\pi {r^2}h{\text{ }} = {\text{ }}\dfrac{1}{3}\pi *{(2)^2}*12{\text{ }} = {\text{ }}16\pi {\text{ }}$ (2)
We are also given with the radius of cone and we know that,
$ \Rightarrow $Radius of cone $ = $ Radius of hemisphere.
As, hemisphere is on the top of the cone.
$ \Rightarrow $So, radius of hemisphere $ = {\text{ r }} = {\text{ }}2{\text{cm}}$
$ \Rightarrow $Volume of hemisphere $ = \dfrac{2}{3}*\pi *{(r)^3}{\text{ }} = {\text{ }}\dfrac{2}{3}*\pi *{(2)^3} = \dfrac{{16}}{3}\pi $ (3)
And we know that,
$ \Rightarrow $Ice-cream in one cone $ = $ volume of cone $ + $ Volume of hemisphere.
So, from equation 2 and 3. We get,
$ \Rightarrow $Ice-cream in one cone $ = 16\pi + \dfrac{{16}}{3}\pi = \dfrac{{64}}{3}\pi $
$ \Rightarrow $And number of cones that can be filled with ice-cream $ = \dfrac{{{\text{Ice - cream inside the cylinder}}}}{{{\text{Ice - cream in one cone}}}}$
$ \Rightarrow $So, number of cones that can be filled with ice-cream $ = {\text{ }}\dfrac{{\pi *{{(8)}^2}*40}}{{\dfrac{{64}}{3}\pi }}{\text{ }} = 120$
$ \Rightarrow $Hence, 120 cones can be filled with Ice-cream.


Note:- Whenever we come up with this type of problem then first find the volume of the cone and then find the volume of the hemisphere. After that add both volumes to get volume of one ice-cream.And we had to divide the volume of one ice-cream by the given volume, to get the number of ice-cream cones that could be made.