Question

An ice-cream cone with a depth of 6 inches and a radius of 1 inch is completely filled with ice cream. If the amount of ice cream in the cone is to be stored in a cylindrical vessel of the same depth and radius, what should the volume of the cylindrical vessel be? Will the ice cream overflow if it melts in the cylindrical vessel?

Answer 1

First, let's draw the ice cream cone.

Given,
The radius of the ice-cream cone \(r=1\) in
The depth of the ice-cream cone \(h=6\) in

As we know,
the volume of a cone \(=\dfrac{1}{3}\pi r^2h\)
\(=\dfrac{1}{3}\pi ×1^2×6\)
\(=\pi×1×2=2 \pi\) cu in

Now, let’s draw the cylindrical vessel of given depth and radius.

Given:
The radius of the cylinder, \(r=1\) in
The depth of the cylinder,\( h=6\) in

As we know,
the volume of the cylinder \(=\pi r^2h\)
\(=\pi ×1^2 ×6=6\pi\) cu in

Concept:

• If the volume of a cone = volume of the cylinder, the cylindrical vessel will be completely filled with melted ice cream.
• If the volume of the cone < volume of the cylinder, the cylindrical vessel will have some vacant space even after the ice cream melts.
• If the volume of the cone > volume of the cylinder, the melted ice cream will overflow from the cylindrical vessel.

The volume of a cone is less than the volume of a cylinder. Hence, the cylindrical vessel will have some vacant space after the ice cream has melted.