Question

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?

Answer 1

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

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