Question

Cups A and B are cone shaped and have heights of `32 cm` and `16 cm` and openings with radii of `15 cm` and `12 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

`4.76cm" "`high from the bottom of the cup

Explanation:

`V_("cone")=1/3pir^2h`

where`" "r="the base radius of the cone ",`

`"and "h="the perpendicular height of the cone"`

we have cups a & B

Cup A`" "r=15cm,h=32cm`

Cup B`" "r=12cm, h=16cm`

first we find the two volumes

`V_A=1/3pi15^2xx32`

`V_A=2400picm^3`

`V_B=1/3pi12^2xx16`
`V_B=768picm^3`

now cup B is full and poured into cup A

we see

`V_B < V_A`

so cup A will not overflow

how high up will A be filled.

using the volume formula once more

`V_(A')=1/3pir^2h`

where `h` is the height from the base of the cone

that is from the top of the cup

`768cancel(pi)=1/3cancel(pi)15^2h`

`h=(768xx3)/15^2=10.24cm`

from the bottom of the cup

`H=15-10.24=4.76cm`