# Cups A and B are cone shaped and have heights of 32 cm and 12 cm and openings with radii of 18 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?

Mar 4, 2016

Find the volume of each one and compare them. Then, use cup's A volume on cup B and find the height.

Cup A will not overflow and height will be:

${h}_{A} ' = 1 , \overline{333} c m$

#### Explanation:

The volume of a cone:

$V = \frac{1}{3} b \cdot h$

where $b$ is the base and equal to π*r^2
$h$ is the height.

Cup A

${V}_{A} = \frac{1}{3} {b}_{A} \cdot {h}_{A}$

V_A=1/3(π*18^2)*32

V_A=3456πcm^3

Cup B

${V}_{B} = \frac{1}{3} {b}_{B} \cdot {h}_{B}$

V_B=1/3(π*6^2)*12

V_B=144πcm^3

Since ${V}_{A} > {V}_{B}$ the cup will not overflow. The new liquid volume of cup A after the pouring will be ${V}_{A} ' = {V}_{B}$:

${V}_{A} ' = \frac{1}{3} {b}_{A} \cdot {h}_{A} '$

${V}_{B} = \frac{1}{3} {b}_{A} \cdot {h}_{A} '$

${h}_{A} ' = 3 \frac{{V}_{B}}{b} _ A$

h_A'=3(144π)/(π*18^2)

${h}_{A} ' = 1 , \overline{333} c m$