# Cups A and B are cone shaped and have heights of 24 cm and 23 cm and openings with radii of 11 cm and 9 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?

Mar 12, 2016

$\approx 20.7 c m$

#### Explanation:

Volume of a cone is given by $\frac{1}{3} \pi {r}^{2} h$, hence

Volume of cone A is $\frac{1}{3} \pi {11}^{2} \cdot 24 = 8 \cdot {11}^{2} \pi = 968 \pi$ and

Volume of cone B is $\frac{1}{3} \pi {9}^{2} \cdot 23 = 27 \cdot 23 \pi = 621 \pi$

It is obvious that when contents of a full cone B are poured into cone A, it will not overflow. Let it reach where upper circular surface will form a circle of radius $x$ and will reach a height of $y$,
then the relation becomes
$\frac{x}{11} = \frac{y}{24} \implies x = \frac{11 y}{24}$
So equating $\frac{1}{3} \pi {x}^{2} y = 621 \pi$
$\implies \frac{1}{3} \pi {\left(\frac{11 y}{24}\right)}^{2} y = 621 \pi$
$\implies {y}^{3} = \frac{621 \cdot 3 \cdot {24}^{2}}{11} ^ 2 \approx 20.7 c m$