Cups A and B are cone shaped and have heights of #25 cm# and #26 cm# and openings with radii of #9 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?

1 Answer
Jun 14, 2018

See below for explanation

Explanation:

Cone Volume is given by #V=1/3pir^2h#

We have cone #A# which volume is #V_A=1/3pi9^2·25=675pi#

For cone #B#: #V_B=1/3pi7^2·26=1274/3pi#

Comparing both volume it is obvius that #1274/3<675# then the cup B do not overfilled cup A

In this case, what will be the height of cup A filled?. Lets see.

With volume B into cone A, we have a situation like this
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By Thales theorem we know that #9/r=25/h# then #h=(25r)/9# (1)

Revolving DEC around CA axis, we have the cone produced by volume B in cup A, so his volume is known. Then

#1274/3pi=1/3pir^2·25r/9# we can remove #pi# and #1/3#

#1274=25r^3/9# from here we have #r^3=1274/25·9=458.64#

Then #r=root(3)458.64=7.71182# cm

In (1), we have #h=25·7.71182/9=21.42# cm