Question

Cups A and B are cone shaped and have heights of `33 cm` and `27 cm` and openings with radii of `13 cm` and `8 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

Cup A will not overflow because `V_b < V_a`
Cup A will be filled up to `h=22.3303` cm

Explanation:

Compute for the volumes of cups A and B first.
The formula for cone

`V=1/3*pi r^2 h`

For cup A
`V_a=1/3*pi*13^2*33=1859pi=5840.220741927`

For cup B
`V_b=1/3*pi*8^2*27=576pi=1809.557368128`

We need 2 equations to solve for unknown height with `V_b` poured into `V_a`

For cup A, we need the ratio of Radius to Height:
`r_a/h_a=13/33`
and `r_a=13/33h_a`

Using volume of `V_b=576pi`

`V_b=1/3pi *r_a^2*h_a`

`576pi=1/3*pi*(13/33*h_a)^2*h_a`

`(576*3*33^2)/13^2=h_a^3`

`h_a=root3((576*3*33^2)/13^2`

`h_a=22.3303" " "`cm

God bless...I hope the explanation is useful.