# Cups A and B are cone shaped and have heights of 33 cm and 37 cm and openings with radii of 12 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?

Sep 17, 2017

No, it will only be filled ~~73.4% of the way

#### Explanation:

The formula for volume of a cone is $V = \pi {r}^{2} \frac{h}{3}$, where $r$ is the radius and $h$ is the height of the cone. For Cup A, $r = 33$ and $h = 12$. Plug that into the formula and you get
$V = \pi {33}^{2} \cdot \frac{12}{3}$
$V = 1089 \pi \cdot 4$
$V = 4356 \pi$

For Cup B, $r = 37$ and $h = 7$. Plug that into the formula and you get
$V = \pi {37}^{2} \cdot \frac{7}{3}$
$V = 1369 \pi \cdot \frac{7}{3}$
$V = \frac{9583 \pi}{3}$
$V \approx 3194.3 \pi$

As you can see, Cup A has a bigger volume then Cup B, since $4356 > 3194.3$, therefor Cup A will not overflow when Cup B is pored into it. As for how high the contents will reach in Cup A, you simply divide Cup B's volume by Cup A's volume to get
(4356pi)/(3194.3pi)=9583/13068~~73.4%

I hope I helped!