a racing track whose left and right ends are semicircular . the distance between two inner parallel line segments is 70 m and they are each 105 m long . if the track is 7 m wide , find the difference in the lengths of the inner edge and outer edge of the track .
The inner rectangle is 70x105
the inner circle has radius 35
the outer rectangle is 84x105
the outer circle has radius 42
now just figure the areas and subtract inner from outer
To find the difference in lengths between the inner and outer edges of the track, we need to calculate the lengths of both edges first.
Let's start by calculating the length of the inner edge of the track.
The inner edge consists of two semicircles and two straight segments.
Each semicircle has a radius equal to half the width of the track, which is 7 m / 2 = 3.5 m.
The formula to calculate the length of a semicircle is L = πr, where L is the length and r is the radius.
Therefore, the length of each semicircle is L = π × 3.5 m = 10.9956 m.
The length of each straight segment is given as 105 m.
So, the total length of the inner edge is 2 × (length of semicircle) + 2 × (length of straight segment) = 2 × 10.9956 m + 2 × 105 m = 21.9912 m + 210 m = 231.9912 m.
Now, let's calculate the length of the outer edge of the track.
The outer edge consists of two semicircles with a larger radius.
The larger radius is equal to the sum of the width of the track and the radius of the inner semicircle.
Therefore, the larger radius is 7 m + 3.5 m = 10.5 m.
Using the same formula, the length of each semicircle is L = π × 10.5 m = 32.9867 m.
The total length of the outer edge is 2 × (length of semicircle) = 2 × 32.9867 m = 65.9734 m.
Finally, let's find the difference between the lengths of the inner and outer edges of the track.
Difference = Length of the outer edge - Length of the inner edge = 65.9734 m - 231.9912 m = -166.0178 m.
Therefore, the difference in lengths between the inner and outer edges of the track is -166.0178 meters.
To find the difference in lengths between the inner and outer edges of the track, we need to calculate the lengths of both edges.
Let's break down the problem into smaller parts.
1. Calculate the length of one semicircle:
- The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius.
- Since we have a semicircle, we divide the circumference by 2.
- The radius of the semicircle is half the width of the track, which is 7/2 = 3.5 m.
- Therefore, the length of one semicircle is (2π * 3.5) / 2 = 3.5π m.
2. Calculate the lengths of the inner and outer edges:
- The track consists of two straight line segments and two semicircles.
- The length of each straight line segment is given as 105 m.
- The length of each semicircle is 3.5π m (as calculated in the previous step).
- There are two semicircles in total, one at each end of the track.
- So, the length of the inner edge is 2(105 + 3.5π) m.
- The distance between the two inner parallel line segments is given as 70 m, which is the width of the track.
- Therefore, the length of the outer edge can be obtained by adding the width to the length of the inner edge: 2(105 + 3.5π) + 70 = 2(175 + 3.5π) m.
3. Find the difference in lengths between the inner and outer edges:
- Subtract the length of the inner edge from the length of the outer edge: 2(175 + 3.5π) - 2(105 + 3.5π) = 350 + 7π - 210 - 7π = 140 m.
Therefore, the difference in lengths between the inner and outer edges of the track is 140 m.