Difference between revisions of "1953 AHSME Problems/Problem 11"
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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>. | 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) } | + | The difference between the circumferences is <math>2\pi{r}+20\pi-2\pi{r}=20\pi\approx20\cdot3=\boxed{\textbf{(C) } 5\text{ feet}}</math>. |
==See Also== | ==See Also== | ||
Revision as of 00:27, 9 April 2020
A running track is the ring formed by two concentric circles. If the circumferences of the two circles differ by 
feet, how wide is the track in feet?
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 
feet larger than the radius of the smaller circle.
Since the circumference of a circle is calculated as 
where 
is the radius, we know that the circumference of the smaller circle is and the circumference of the larger circle is 
.
The difference between the circumferences is 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
