Question

Cups A and B are cone shaped and have heights of `32 cm` and `12 cm` and openings with radii of `18 cm` and `6 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

Find the volume of each one and compare them. Then, use cup's A volume on cup B and find the height.

Cup A will not overflow and height will be:

`h_A'=1,bar(333)cm`

Explanation:

The volume of a cone:

`V=1/3b*h`

where `b` is the base and equal to `π*r^2`
`h` is the height.

Cup A

`V_A=1/3b_A*h_A`

`V_A=1/3(π*18^2)*32`

`V_A=3456πcm^3`

Cup B

`V_B=1/3b_B*h_B`

`V_B=1/3(π*6^2)*12`

`V_B=144πcm^3`

Since `V_A>V_B` the cup will not overflow. The new liquid volume of cup A after the pouring will be `V_A'=V_B`:

`V_A'=1/3b_A*h_A'`

`V_B=1/3b_A*h_A'`

`h_A'=3(V_B)/b_A`

`h_A'=3(144π)/(π*18^2)`

`h_A'=1,bar(333)cm`