# Cups A and B are cone shaped and have heights of 22 cm and 23 cm and openings with radii of 6 cm and 12 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?

Feb 22, 2016

$\textcolor{b r o w n}{\text{Cone B has both a larger base diameter and greater height.}}$
$\textcolor{b r o w n}{\text{So it is evident that it must have the larger volume. In}}$$\textcolor{b r o w n}{\text{conclusion, cup A will overflow.}}$

#### Explanation:

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$\textcolor{b l u e}{\text{Difference in volume demonstrated}}$

Volume of a cone is $\frac{1}{3} \text{ base area "xx" height}$

Cone A base area$\text{ " pir^2" "->" "pi6^2" "->" } 36 \pi$
Cone A height 22cm
$\implies \text{Volume cone A =} 36 \pi \times 22 = 792 \pi c {m}^{3}$
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Cone B base area$\text{ " pir^2" "->" "pi12^2" "->" } 144 \pi$
Cone B height 23cm
$\implies \text{Volume cone B =} 144 \pi \times 23 = 3312 \pi c {m}^{3}$
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Clearly cup B has a significantly larger volume.

So as the question posed stands cup A will overflow.

$\textcolor{g r e e n}{\text{However, my experience of questions leads me to ask; should it be}}$ $\textcolor{g r e e n}{\text{cup A that is bigger? Otherwise there is no point in even}}$$\textcolor{g r e e n}{\text{mentioning about height of fill level.}}$