What are we learning ?

Competency in Focus:Geometry of circles. This problem from American Mathematics contest (AMC 8, 2014) is based on simple counting of semicircles.

First look at the knowledge graph.

Next understand the 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?
Note: 1 mile = 5280feet amc 8 2014 problem no 25
Source of the problem
American Mathematical Contest 2014, AMC 8 Problem 25
Key Competency
Geometry of circles
Difficulty Level
4/10
Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

Start with hints

Do you really need a hint ? Try it first!
How many lanes the highway consists of ? 2 right! given highway is 40 feet wide .Then width of each lane will be 40/2=20 feet wide .
Look at the diagram .See that the radius of each semicircle will be 20 feet on which Robert must be riding his bike .Again see that each semicrcle covers 40 feet of highway i.e. the diameter of the semicircle .
Calculate the number of semicircles over the whole mile . Number of semicircles=(length of the highway covered in total by Robert)/(length of highway covered by each semicircle)=5280/40 [since 1 mile=5280 feet] =132.
Where the semicircles full circles ,their circumference would be 2.\( \pi \) r =2.\( \pi \).20=40 \( \pi \) feet (since r=radius=20 feet). Therefore the circumference of semicircles is half that, or 20.\( \pi \) feet.
Therefore over the stretch of hghway, Robert rides a total of 132.20.\( \pi \)=2640. \( \pi \) feet equivalent to \(\frac{ \pi}{2} \) mile( since 1 mile=5280 feet) Given Robert rides at 5 miles per hour.So, time required by Robert =distance travelled/rate=(\( \frac{\pi}{2} \) miles)/(5 miles per hour)= \( \frac{\pi}{10} \)hours.  

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