# Cups A and B are cone shaped and have heights of #33 cm# and #37 cm# and openings with radii of #10 cm# and #7 cm#, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

##### 2 Answers

#### Explanation:

Consider across section vertically through the centre of each cone.

If the area of the taller cross section will fit in the area of the shorter one then so will the volumes.

Let cross section area be

we require that

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Volume a circular cone is

Thus the volume of B is:

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The radius will vary in accordance with the height from zero to full radius at the top height of 33cm.

So

The volume of the transferred material is

So

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

height =

#### Explanation:

We need to find and compare the Volumes of A and B.

Check first whether they are similar in shape - this would make some of the calculations easier.

Are the sides in the same ratio?

Which is bigger? We do not need to include

Therefore A will not overflow but we need to find the height.

The cone formed by the water in A and the whole cone of A are similar in shape.

The ratio of the cubes of the heights is equal to the ratio of the volumes.