Difference between revisions of "1953 AHSME Problems/Problem 11"
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== 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 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>. | + | 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. |
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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>. | ||
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| + | 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>. | ||
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| + | ==See Also== | ||
| + | |||
| + | {{AHSME 50p box|year=1953|num-b=10|num-a=11}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 19:48, 1 April 2017
Problem
A running track is the ring formed by two concentric circles. It is 
feet wide. The circumference of the two circles differ by about:
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 11 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
