Question

An ice cream ball of diameter $6\mathrm{cm}$. is placed over a cone of radius$2.5\mathrm{cm}$. and height $10\mathrm{cm}$.

Is the cone big enough to hold all the ice cream if it melts?

Answer 1

Step-1: Check when the ice cream melts, will the cone be large enough to hold it all.

A scoop of ice cream with a diameter of $6\mathrm{cm}$.

The Ice cream cone with radius $2.5cm$ and height $10\mathrm{cm}$.

It's known that,

The volume $V$of a sphere with radius $r$ is $V=\frac{4}{3}\pi r^3$and the diameter of the sphere is $6cm$ (This implies that the radius is $3cm$.).

Thus,

$\begin{array}{rcl}V& =& \frac{4}{3}\pi \times 3^3\\ & =& 36\pi \end{array}$

Step-2: Find the volume of a cone:
The volume $V$of a cone of radius $r$ and height $\left(h\right)$ is $V=\frac{1}{3}\pi r^{2}h$.
The diameter of the cone is $5cm$. i.e. radius is $2.5cm$. and its height is $10cm$.

Then,

$\begin{array}{rcl}V& =& \frac{1}{3}\pi \times 2.5^2\times 10\\ & =& \frac{62.5}{3}\pi \\ & =& 20.8333\pi \end{array}$

The difference between the volume of an ice cream scoop and volume of cone is,

$$\begin{gathered} =36\mathrm{\pi }-20.833\mathrm{\pi } \\ =16.167\mathrm{\pi } \end{gathered}$$
The volume of an ice cream scoop is less than $36\mathrm{\pi }$.

Therefore, the cone is not big enough to hold all the ice cream if it melts.