Difference between revisions of "2014 AMC 8 Problems/Problem 25"
m (→Problem) |
Murmillo15 (talk | contribs) |
||
| (54 intermediate revisions by 35 users not shown) | |||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
| − | A straight one-mile stretch of highway, | + | A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch? |
| − | + | <asy>size(10cm); pathpen=black; pointpen=black; | |
| − | |||
| − | |||
| − | <asy> | ||
| − | size(10cm); pathpen=black; pointpen=black; | ||
D(arc((-2,0),1,300,360)); | D(arc((-2,0),1,300,360)); | ||
D(arc((0,0),1,0,180)); | D(arc((0,0),1,0,180)); | ||
| Line 11: | Line 7: | ||
D(arc((4,0),1,0,180)); | D(arc((4,0),1,0,180)); | ||
D(arc((6,0),1,180,240)); | D(arc((6,0),1,180,240)); | ||
| − | + | D((-1.5,-1)--(5.5,-1));</asy> | |
| − | + | Note: 1 mile = 5280 feet | |
| − | D((-1.5,-1)--(5.5,-1)); | ||
| − | </asy> | ||
<math> \textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad\textbf{(E) }\frac{2\pi}{3} </math> | <math> \textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad\textbf{(E) }\frac{2\pi}{3} </math> | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/zi01koyAVhM ~savannahsolver | ||
==Solution== | ==Solution== | ||
| − | There are two possible interpretations of the problem: that the road as a whole is <math>40</math> feet wide, or that each lane is <math>40</math> feet wide. Both interpretations will arrive at the same result. However, let us stick with the first interpretation for simplicity. Each lane must then be <math>20</math> feet wide, so Robert must be riding his bike in semicircles with radius <math>20</math> feet and diameter <math>40</math> feet. Since the road is <math>5280</math> feet long, over the whole mile, Robert rides <math>\frac{5280}{40} =132</math> semicircles in total. Were the semicircles full circles, their circumference would be <math>2\pi\cdot 20=40\pi</math> feet; as it is, the circumference of each is half that, or <math>20\pi</math> feet. Therefore, over the stretch of highway, Robert rides a total of <math>132\cdot 20\pi =2640\pi</math> feet, equivalent to <math>\frac{\pi}{2}</math> miles. Robert rides at 5 miles per hour, so divide the <math>\frac{\pi}{2}</math> miles by <math>5</math> mph to arrive at <math>\textbf{(B) }\frac{\pi}{10}</math> hours. | + | ===Solution 1=== |
| + | There are two possible interpretations of the problem: that the road as a whole is <math>40</math> feet wide, or that each lane is <math>40</math> feet wide. Both interpretations will arrive at the same result. However, let us stick with the first interpretation for simplicity. Each lane must then be <math>20</math> feet wide, so Robert must be riding his bike in semicircles with radius <math>20</math> feet and diameter <math>40</math> feet. Since the road is <math>5280</math> feet long, over the whole mile, Robert rides <math>\frac{5280}{40} =132</math> semicircles in total. Were the semicircles full circles, their circumference would be <math>2\pi\cdot 20=40\pi</math> feet; as it is, the circumference of each is half that, or <math>20\pi</math> feet. Therefore, over the stretch of highway, Robert rides a total of <math>132\cdot 20\pi =2640\pi</math> feet, equivalent to <math>\frac{\pi}{2}</math> miles. Robert rides at 5 miles per hour, so divide the <math>\frac{\pi}{2}</math> miles by <math>5</math> mph (because <math>t = \frac{d}{r}</math> and time = distance/rate) to arrive at <math>\boxed{\textbf{(B) }\frac{\pi}{10}}</math> hours. | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2014|num-b=24|after=Last Problem}} | {{AMC8 box|year=2014|num-b=24|after=Last Problem}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 19:04, 3 October 2024
Problem
A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?
![[asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,-1)--(5.5,-1));[/asy]](http://latex.artofproblemsolving.com/7/1/a/71a0b57907e6bd351febe1d78fad2839d5131421.png)
Note: 1 mile = 5280 feet
Video Solution
https://youtu.be/zi01koyAVhM ~savannahsolver
Solution
Solution 1
There are two possible interpretations of the problem: that the road as a whole is 
feet wide, or that each lane is feet wide. Both interpretations will arrive at the same result. However, let us stick with the first interpretation for simplicity. Each lane must then be 
feet wide, so Robert must be riding his bike in semicircles with radius feet and diameter feet. Since the road is 
feet long, over the whole mile, Robert rides 
semicircles in total. Were the semicircles full circles, their circumference would be 
feet; as it is, the circumference of each is half that, or 
feet. Therefore, over the stretch of highway, Robert rides a total of 
feet, equivalent to 
miles. Robert rides at 5 miles per hour, so divide the miles by 
mph (because 
and time = distance/rate) to arrive at 
hours.
See Also
| 2014 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last Problem | |
| 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 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
