# Cups A and B are cone shaped and have heights of 32 cm and 14 cm and openings with radii of 15 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?

Aug 24, 2016

height = $20.93 c m$

#### Explanation:

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

${V}_{A} = \frac{\pi \times \textcolor{red}{{15}^{2} \times 32}}{3} \mathmr{and} {V}_{B} = \frac{\pi \times \textcolor{b l u e}{{12}^{2} \times 14}}{3}$

We only need to know which one is bigger, not the actual volumes.
We can therefore ignore $\pi \mathmr{and} 3$ as they are common to both.

By inspection we can see that :$\textcolor{red}{{V}_{A}} > \textcolor{b l u e}{{V}_{B}}$

So A will not overflow, but how high will the water reach?

The water in A and the whole cup A are similar figures.
Therefore we can compare the ratio of the cubes of the heights with the ratio of their volumes.

${\left(\frac{h}{H}\right)}^{3} = {V}_{B} / {V}_{A}$

${\left(\frac{h}{H}\right)}^{3} = {h}^{3} / {H}^{3} = {h}^{3} / {32}^{3} = \frac{\textcolor{b l u e}{{12}^{2} \times 14}}{\textcolor{red}{{15}^{2} \times 32}} = {V}_{B} / {V}_{A}$

${h}^{3} = \frac{{32}^{3} \times {12}^{2} \times 14}{{15}^{2} \times 32} = 9 , 1750.4$

$h = \sqrt[3]{9 , 1750.4} = 20.93 c m$

Dec 17, 2016

Cup A's height will be 7.68cm

#### Explanation:

Volume of cone$= \frac{1}{3} \pi {r}^{2} h$
A$= \frac{1}{3} {\pi}^{2} \cdot {15}^{2} \cdot 32$
$= \frac{1}{3} \cdot 3.141592654 \cdot 225 \cdot 32$
$= \frac{1}{3} \cdot 22619.467$
$= \frac{22619.467}{3}$
Vol. A$= 7539.822 c {m}^{3}$
Vol.B$= \frac{1}{3} \pi {12}^{2} \cdot 12$
$= \frac{1}{3} \cdot 3.41592654 \cdot 144 \cdot 12$
$= \frac{1}{3} \cdot 5428.672105$
$= \frac{5428.672105}{3}$
$V o l . B = 1809.557 c {m}^{3}$
New vol cup A=$\frac{1}{3} \cdot \pi {15}^{2} \cdot x = 1809.557 c {m}^{3}$
$235.619 x = 1809.557$
$x = \frac{1809.557}{235.619}$
$x = 7.68 c m$

Dec 30, 2016

a. not overflow
b. height of cup A = 8.96cm

#### Explanation:

$V o l u m e o f c o n = \frac{1}{3} \cdot \pi \cdot {r}^{2} \cdot h$.
where r = radius and h = height

$V a = \frac{1}{3} \cdot \pi \cdot {15}^{2} \cdot 32$
$V a = 2400 \pi$

$V b = \frac{1}{3} \cdot \pi \cdot {12}^{2} \cdot 14$
$V b = 672 \pi$

so, cup A wont overflow because it is bigger than cup B.

let say T is the height of cup A when content in cup B poured to cup A.

Va = Vb

$\cancel{\frac{1}{3}} \cdot \cancel{\pi} \cdot {15}^{2} \cdot T = \cancel{\frac{1}{3}} \cdot \cancel{\pi} \cdot {12}^{2} \cdot 14 = \left(672 \cdot \pi\right)$

${15}^{2} \cdot T = {12}^{2} \cdot 14$

$T = \frac{{12}^{2} \cdot 14}{{15}^{2}}$

$T = 8.96 c m$