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

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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?
  
(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} \qquad \text{(B)} \sqrt{\frac{216}{\pi}} \qquad \text{(C)} \sqrt{\frac{2160}{\pi}} \qquad \text{(D)} \frac{216}{\pi} \qquad \text{(E)} \frac{2160}{\pi}</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 15:43, 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?

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

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 (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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