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1953 AHSME Problems/Problem 11: Difference between revisions

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A running track is the ring formed by two concentric circles. If the circumferences of the two circles differ by <math>10\pi </math> feet, how wide is the track in feet?
A running track is the ring formed by two concentric circles. It is <math>10</math> feet wide. The circumference of the two circles differ by about:
<math>\textbf{(A)}\ 10\text{ feet} \qquad
\textbf{(B)}\ 30\text{ feet} \qquad
\textbf{(C)}\ 60\text{ feet} \qquad
\textbf{(D)}\ 100\text{ feet}\\ \textbf{(E)}\ \text{none of these}    </math>


== Solution ==
== Solution ==


We notice that since the running track is simply the area of the outer circle that is outside of the inner circle, the radius of the larger circle must be precisely <math>10</math> feet larger than the radius of the smaller circle.  
Since the track is 10 feet wide, the diameter of the outer circle will be 20 feet more than the inner circle. Since the circumference of a circle is directly proportional to its diameter, the difference in the circles' diameters is simply <math>20\pi </math> feet. Using <math>\pi \approx 3</math>, the answer is <math>\fbox{C}</math>.
 
Since the circumference of a circle is calculated as <math>2\pi{r}</math> where <math>r</math> is the radius, we know that the circumference of the smaller circle is <math>2\pi{r}</math> and the circumference of the larger circle is <math>2\pi(r+10)=2\pi{r}+20\pi</math>.
 
The difference between the circumferences is <math>2\pi{r}+20\pi-2\pi{r}=20\pi\approx20\cdot3=\boxed{\textbf{(C) } 60\text{ feet}}</math>.
 


==See Also==
==See Also==

Latest revision as of 11:23, 22 April 2020

A running track is the ring formed by two concentric circles. It is $10$ feet wide. The circumference of the two circles differ by about: $\textbf{(A)}\ 10\text{ feet} \qquad \textbf{(B)}\ 30\text{ feet} \qquad \textbf{(C)}\ 60\text{ feet} \qquad \textbf{(D)}\ 100\text{ feet}\\ \textbf{(E)}\ \text{none of these}$

Solution

Since the track is 10 feet wide, the diameter of the outer circle will be 20 feet more than the inner circle. Since the circumference of a circle is directly proportional to its diameter, the difference in the circles' diameters is simply $20\pi$ feet. Using $\pi \approx 3$, the answer is $\fbox{C}$.

See Also

1953 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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All AHSME Problems and Solutions

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