# 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?

Apr 21, 2018

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.

$A = \setminus \pi {\left(5\right)}^{2} \left(32\right) = \left(25\right) \left(4\right) \left(8\right) \pi = 800 \pi$

$B = \pi {\left(6\right)}^{2} \left(12\right) = \left(36\right) \left(12\right) \pi = 432 \pi$

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 $h$:

$h = \frac{V}{\setminus \pi {r}^{2}} = \setminus \frac{432 \pi}{\pi \left({5}^{2}\right)} = \setminus \frac{432}{25} = 17.28$ cm

Apr 21, 2018

$\textcolor{b l u e}{17.28 c m}$

#### Explanation:

Volume of a cone is given by:

$V = \frac{1}{3} \pi {r}^{2} h$

First find the volumes of A and B:

Volume of A:

$V = \frac{1}{3} \pi {\left(5\right)}^{2} \left(32\right) = \frac{800 \pi}{3}$

Volume of B:

$V = \frac{1}{3} \pi {\left(6\right)}^{2} \left(12\right) = 144 \pi$

So the contents of B will no overflow when poured into A. To find the height it will reach we solve for $\boldsymbol{h}$.

$V = \frac{1}{3} \pi {r}^{2} h$

$h = \frac{3 V}{\pi {r}^{2}}$

$h = \frac{3 \left(144 \pi\right)}{\pi {\left(5\right)}^{2}} = \frac{432}{{\left(5\right)}^{2}} = \frac{432}{25} = 17.28$cm