Question

A cylindrical container is filled with ice-cream, whose diameter is 12cm and height is 15cm. The whole ice-cream is distributed to 10 children in equal cones having hemispherical tops. If the height of the conical portion is twice the diameter of its base, find the diameter of the ice-cream.

Answer 1

Hint: In this problem, the volume of the 1 ice-cream cone is obtained by dividing the volume of the container by 10. Now, apply the formula for the volume of the cone with hemispherical tops to find the diameter of the cone.

Complete step by step solution:
The formula for the volume V of a cylinder having diameter d and height h is shown below.
\[
  \,\,\,\,\,\,V = \pi {r^2}h \\
   \Rightarrow V = \dfrac{\pi }{4}{d^2}h\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {r = \dfrac{d}{2}} \right) \\
\]
Now, substitute 12 for d and 15 for h in the above equation, to obtain the volume of the cylindrical container.
\[
  \,\,\,\,\,V = \dfrac{\pi }{4}{\left( {12} \right)^2}\left( {15} \right) \\
   \Rightarrow V = \dfrac{\pi }{4}\left( {144} \right)\left( {15} \right) \\
   \Rightarrow V = \pi \left( {36} \right)\left( {15} \right) \\
   \Rightarrow V = 540\pi \\
\]
Since, whole ice-cream is distributed to 10 children, the volume of one ice-cream cone is obtained as follows:
\[
  {\text{Volume of 10 cone}} \to 540\pi \\
  {\text{Volume of 1 cone}} \to 54\pi \\
\]
The formula for the volume of 1 cone is shown below.
\[\dfrac{\pi }{{12}}{D^3} + \dfrac{1}{{12}}\pi {D^2}H = 54\pi\]
Since, the height of the conical portion is twice the diameter of its base, substitute 2D for H in the above formula.
\[
  \,\,\,\,\,\,\dfrac{\pi }{{12}}{D^3} + \dfrac{1}{{12}}\pi {D^2}\left( {2D} \right) = 54\pi \\
   \Rightarrow \dfrac{\pi }{{12}}{D^3} + \dfrac{2}{{12}}\pi {D^3} = 54\pi \\
   \Rightarrow \dfrac{{3\pi }}{{12}}{D^3} = 54\pi \\
   \Rightarrow \dfrac{1}{4}{D^3} = 54 \\
   \Rightarrow {D^3} = 216 \\
   \Rightarrow D = 6 \\
\]

Thus, the diameter of the ice-cream cone is 6cm.

Note: In this case, the volume of the cylindrical container is equal to the 10 times the volume of the ice-cream cone. The formula for the volume of the cylinder in terms of diameter is \[V = \dfrac{\pi }{4}{d^2}h\], here d is diameter and h is the height of the cylinder.