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In a city park, there are two paths between points A and B:
1. The “Hill Path” is a single large semicircle with diameter AB.
2. The “Valley Path” consists of three smaller semicircles. The diameters of these semicircles lie on line AB and are of equal length.
The straight-line distance between A and B is 120 meters.
If two friends start at point A at the same time and walk at the same speed, one taking the Hill Path and the other the Valley Path, who will reach point B first, or will they arrive at the same time?
Let’s solve this step-by-step:
1. For a semicircle, the length of the curved path is π * r, where r is the radius.
2. Hill Path:
– Diameter = 120 m
– Radius = 60 m
– Length of Hill Path = π * 60 = 60π meters
3. Valley Path:
– Total length of AB = 120 m
– Each small semicircle has a diameter of 120 / 3 = 40 m
– Radius of each small semicircle = 20 m
– Length of each small semicircle = π * 20 = 20π meters
– Total length of Valley Path = 3 * 20π = 60π meters
4. Comparison:
Length of Hill Path = Length of Valley Path = 60π meters
Therefore, both friends will arrive at point B at exactly the same time, assuming they walk at the same speed.
This result might be surprising! It demonstrates an interesting property of semicircles: the total length of any number of semicircles arranged in this way will always be equal to the length of a single semicircle with a diameter equal to the total straight-line distance.