Question

A scoop of ice cream in the shape of a whole sphere sits in a right cone. the radius of the ice cream scoop is 2.5 cm and the radius of the cone is 2.5 cm. How tall must the height of the cone be to fit all the ice cream without spilling if it melts? SHOW ALL YOUR WORK!!!

Answer 1

Answer:

(At least) 10 centimeters.

Step-by-step explanation:

The radius of the ice cream scoop is 2.5 cm and the radius of the cone is also 2.5 cm.

We want to determine the height of the cone such that it will fit all of the ice cream when it melts without any spilling.

First, we will find the volume of the ice cream scoop. The volume for a sphere is given by:

$\displaystyle V=\frac{4}{3}\pi r^3$

Since the radius is 2.5 cm, the volume of the full ice cream scoop is:

$\displaystyle V=\frac{4}{3}\pi (2.5)^3$

Use a calculator:

$V\approx 65.4498\text{ cm}^3$

The volume of a cone is given by:

$\displaystyle V=\frac{1}{3}\pi r^2h$

The radius of the cone is 2.5 cm. Therefore:

$\displaystyle V=\frac{1}{3}\pi(2.5)^2h=\frac{1}{3}\pi(6.25 h)=\frac{6.25\pi}{3}h$

The volume of the cone should be equal to the volume of the scoop. So:

$\displaystyle 65.4498=\frac{6.25\pi}{3}h$

Solve for h:

$\displaystyle h\approx 65.4498\Big(\frac{3}{6.25\pi}\Big)=10\text{ cm}$

The height of the cone should be (at least) 10 cm.