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Difference between revisions of "1991 AHSME Problems/Problem 8"

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== Problem ==
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Liquid <math>X</math> does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film <math>0.1</math>cm thick. A rectangular box measuring <math>6</math>cm by <math>3</math>cm by <math>12</math>cm is filled with liquid <math>X</math>. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?
 
Liquid <math>X</math> does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film <math>0.1</math>cm thick. A rectangular box measuring <math>6</math>cm by <math>3</math>cm by <math>12</math>cm is filled with liquid <math>X</math>. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?
  
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(A) <math>\frac{\sqrt{216}}{\pi}</math> (B) <math>\sqrt{\frac{216}{\pi}}</math> (C) <math>\sqrt{\frac{2160}{\pi}}</math> (D) <math>\frac{216}{\pi}</math> (E) <math>\frac{2160}{\pi}</math>
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<math>\text{(A)} \frac{\sqrt{216}}{\pi} \quad \text{(B)} \sqrt{\frac{216}{\pi}} \quad \text{(C)} \sqrt{\frac{2160}{\pi}} \quad \text{(D)} \frac{216}{\pi} \quad \text{(E)} \frac{2160}{\pi}</math>
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== Solution ==
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<math>\fbox{C}</math> The volume of liquid <math>X</math> is <math>6 \times 3 \times 12 = 216</math>, so if the radius is <math>r</math>, then using the formula for the volume of a cylinder, we get <math>\pi r^2 \times 0.1 = 216 \implies \pi r^2 = 2160 \implies r = \sqrt{\frac{2160}{\pi}}.</math>
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== See also ==
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{{AHSME box|year=1991|num-b=7|num-a=9}} 
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[[Category: Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 15:44, 14 March 2023

Problem

Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?

$\text{(A)} \frac{\sqrt{216}}{\pi} \quad \text{(B)} \sqrt{\frac{216}{\pi}} \quad \text{(C)} \sqrt{\frac{2160}{\pi}} \quad \text{(D)} \frac{216}{\pi} \quad \text{(E)} \frac{2160}{\pi}$

Solution

$\fbox{C}$ The volume of liquid $X$ is $6 \times 3 \times 12 = 216$, so if the radius is $r$, then using the formula for the volume of a cylinder, we get $\pi r^2 \times 0.1 = 216 \implies \pi r^2 = 2160 \implies r = \sqrt{\frac{2160}{\pi}}.$

See also

1991 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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