Question

Cups A and B are cone shaped and have heights of `33 cm` and `37 cm` and openings with radii of `10 cm` and `13 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?

Answer 1

`29.9cm`

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?
`13/10 =1.3 and 37/33 =1.12 " "rArr " A and B are not similar"`

`Vol_("cone") = (pi r^2 h)/3`

`Vol_A = (13^2xx37 xxpi)/3" "and " "Vol_B = (10^2xx33xxpi)/3`

By inspection we can see that `Vol A > Vol B`

`color(white)(xxxxxxxxxxxxxxxxx)13^2 xx37 > 10^2 xx33`

Therefore A will not overflow but we need 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.

`color(white)(xxxxxxxxxxxxxxxxx)h^3/H^3 = v/V`

`h^3/37^3 = (10^2 xx33)/(13^2 xx 37)`

`h^3 = (37^3 xx10^2 xx33)/(13^2xx37) = 26,731.95`

`h = root3(26,731.95)`

`29.9cm`