Question

Cups A and B are cone shaped and have heights of `32 cm` and `12 cm` and openings with radii of `5 cm` and `6 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

There's no way the contents of that little cup will overflow that tall glass, and indeed it fills to height of exactly 17.28 cm.

Explanation:

We have the volume of a cup or cylinder of height `h` and radius `r` is `V=pi r^2 h`. The `pi` doesn't particularly matter for this question as we'll see.

`A = \pi (5)^2 (32) = (25)(4)(8) pi = 800 pi`

`B = pi (6)^2 (12) = (36)(12)pi = 432 pi`

So there will be plenty of room in `A` for the contents of `B`. That would probably be pretty obvious to anyone comparing the rather squat cup to the tall glass.

To get the height, we solve for `h`:

`h = frac{V}{\pi r^2} = \frac{432 pi}{pi (5^2)} = \frac{432}[25} = 17.28` cm

Answer 2

`color(blue)(17.28cm)`

Explanation:

Volume of a cone is given by:

`V=1/3pir^2h`

First find the volumes of A and B:

Volume of A:

`V=1/3pi(5)^2(32)=(800pi)/3`

Volume of B:

`V=1/3pi(6)^2(12)=144pi`

So the contents of B will no overflow when poured into A. To find the height it will reach we solve for `bbh`.

`V=1/3pir^2h`

`h=(3V)/(pir^2)`

`h=(3(144pi))/(pi(5)^2)=(432)/((5)^2)=432/25=17.28`cm