Cups A and B are cone shaped and have heights of $33 c m$ $33 cm$ and $37 c m$ $37 cm$ and openings with radii of $10 c m$ $10 cm$ and $13 c m$ $13 cm$ , 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?

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Cups A and B are cone shaped and have heights of $33 c m$ $33 cm$ and $37 c m$ $37 cm$ and openings with radii of $10 c m$ $10 cm$ and $13 c m$ $13 cm$ , 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?

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Answer 1 · 2016-08-23T15:42:58

$29.9 c m$ $29.9cm$

Explanation:

We need to find and compare the Volumes of A and B.
Check first whether they are similar in shape - this would make some of the calculations easier.

Are the sides in the same ratio?
$\frac{13}{10} = 1.3 and \frac{37}{33} = 1.12 \Rightarrow A and B are not similar$ $13/10 =1.3 and 37/33 =1.12 " "rArr " A and B are not similar"$

$V o l_{cone} = \frac{π r^{2} h}{3}$ $Vol_("cone") = (pi r^2 h)/3$

$V o l_{A} = \frac{13^{2} \times 37 \times π}{3} and V o l_{B} = \frac{10^{2} \times 33 \times π}{3}$ $Vol_A = (13^2xx37 xxpi)/3" "and " "Vol_B = (10^2xx33xxpi)/3$

By inspection we can see that $V o l A > V o l B$ $Vol A > Vol B$

$\times \times \times \times \times \times \times \times x 13^{2} \times 37 > 10^{2} \times 33$ $color(white)(xxxxxxxxxxxxxxxxx)13^2 xx37 > 10^2 xx33$

Therefore A will not overflow but we need the height.

The cone formed by the water in A and the whole cone of A are similar in shape.

The ratio of the cubes of the heights is equal to the ratio of the volumes.

$\times \times \times \times \times \times \times \times x \frac{h^{3}}{H^{3}} = \frac{v}{V}$ $color(white)(xxxxxxxxxxxxxxxxx)h^3/H^3 = v/V$

$\frac{h^{3}}{37^{3}} = \frac{10^{2} \times 33}{13^{2} \times 37}$ $h^3/37^3 = (10^2 xx33)/(13^2 xx 37)$

$h^{3} = \frac{37^{3} \times 10^{2} \times 33}{13^{2} \times 37} = 26, 731.95$ $h^3 = (37^3 xx10^2 xx33)/(13^2xx37) = 26,731.95$

$h = \sqrt[3]{26, 731.95}$ $h = root3(26,731.95)$

$29.9 c m$ $29.9cm$

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Cups A and B are cone shaped and have heights of $33 c m$ $33 cm$ and $37 c m$ $37 cm$ and openings with radii of $10 c m$ $10 cm$ and $13 c m$ $13 cm$ , 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?

1 Answer

Explanation:

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Cups A and B are cone shaped and have heights of 33 cm33cm33 cm and 37 cm37cm37 cm and openings with radii of 10 cm10cm10 cm and 13 cm13cm13 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?

1 Answer

Explanation:

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Cups A and B are cone shaped and have heights of $33 c m$ $33 cm$ and $37 c m$ $37 cm$ and openings with radii of $10 c m$ $10 cm$ and $13 c m$ $13 cm$ , 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?