A running track is designed with two semicircular ends and two parallel straightaways. If the inner perimeter of the track is to be exactly 400 meters, what are reasonable dimensions for the radius and straightaway of the track? draw picure, justify answer

What you are looking for is the perimeter of a circle (the two semicircles) and two sides of a rectangle = 400 meters. The diameter of the circle is the length of the other two sides.

I hope this helps.

To find reasonable dimensions for the radius and straightaways of the running track, we can use the given information that the inner perimeter of the track is 400 meters.

Let's start by drawing a diagram. The track consists of two semicircular ends and two parallel straightaways. The semicircular ends are joined by two straightaways.

____________________
| |
r/2 | | r/2
| 400m |
|____________________|
straightaway

Let's consider the perimeter of the track:

Perimeter = Perimeter of the two semicircles + Perimeter of the two straightaways
400m = πr + 2s

Since the straightaways are parallel and equal in length, we can represent their total length as 2s.

Given that the straightaway lengths should be reasonable, we can assume they are not too short or too long. So, let's approximate the lengths of the straightaways to be equal to each other.

Therefore, we have:
400m = πr + 2s

To simplify the equation, we need to find a relationship between the radius (r) and the straightaway length (s).

The length of the straightaway can be calculated using the formula for the circumference of a circle:

Circumference = 2πr

Since the straightaway is equal to half the circumference of the circle it joins, we have:
s = (2πr)/2 = πr

Substituting this value back into the equation:
400m = πr + 2(πr)
400m = 3πr

Now, let's solve for r:
r = (400m)/(3π)

To find reasonable values for the radius and straightaway, we can use an approximation for the value of π. Taking π ≈ 3.14, we can solve for r:

r ≈ (400m)/(3 * 3.14)
r ≈ 42.57 meters

With the radius (r) approximately equal to 42.57 meters, we can calculate the straightaway length (s) using the relationship we found earlier:

s = πr
s ≈ 3.14 * 42.57
s ≈ 133.6 meters

Therefore, reasonable dimensions for the radius and straightaway of the running track are approximately 42.57 meters and 133.6 meters, respectively.