Question

Cups A and B are cone shaped and have heights of `33 cm` and `37 cm` and openings with radii of `12 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?

Answer 1

No, it will only be filled `~~73.4%` of the way

Explanation:

The formula for volume of a cone is `V=pir^2 h/3`, where `r` is the radius and `h` is the height of the cone. For Cup A, `r=33` and `h=12`. Plug that into the formula and you get
`V=pi33^2 * 12/3`
`V=1089pi * 4`
`V=4356pi`

For Cup B, `r=37` and `h=7`. Plug that into the formula and you get
`V=pi37^2 * 7/3`
`V=1369pi *7/3`
`V=(9583pi)/3`
`V~~3194.3pi`

As you can see, Cup A has a bigger volume then Cup B, since `4356>3194.3`, therefor Cup A will not overflow when Cup B is pored into it. As for how high the contents will reach in Cup A, you simply divide Cup B's volume by Cup A's volume to get
`(4356pi)/(3194.3pi)=9583/13068~~73.4%`

I hope I helped!