Question

Cups A and B are cone shaped and have heights of `39 cm` and `25 cm` and openings with radii of `17 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

Cup A will not overflow. Cup A will be filled to a height of 28.12cm

Explanation:

cone's volume formula = `pir^2xxh/3`
h=height, r=radius

Cup A, height 39, radius 17, `vol = 11803.0 cm^3`
Cup B, height 25, radius 13, `vol = 4424.4 cm^3`

As Cup A's volume is larger than Cup B's, Cup A will not overflow.

Now if `4424.4 cm^3` of water is poured into Cup A, then Cup A will be filled to a height of h1, at this height, the radius is, say, r1.

`r/h = (r1)/(h1) => r1=(rh1)/h`
`V=pi(r1)^2(h1)/3`
`=pi(r(h1)/h)^2(h1)/3`
`=pi(r^2(h1)^2/h^2)(h1)/3`
`=(pi/3) ((r^2(h1)^3)/(h^2))`
`(h1)^3= (3Vh^2)/(pir^2)`
`(h1)^3= (3xx4424.4xx39^2)/(pixx17^2) =22236.0`

`h1= 28.12 cm`

`r1= 17xx28.12/39 = 12.25 cm`