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?

1 Answer
Sep 7, 2016

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#