To solve this problem, let's break it down step-by-step:

Problem Description

The track consists of:

1. Two straight sections of 90 meters each.
2. Two semicircular sections combined to form a circular path.

The total length of the track is 400 meters. We need to find the diameter $d$ of the semicircular sections.

Steps to Solve

The total perimeter of the track is: $\text{Total Length} = \text{Straight Sections} + \text{Circular Sections}$ Let $C$ be the circumference of the full circle formed by the semicircles: $\text{Circular Sections} = \frac{C}{2} = \pi \cdot r$ where $r$ is the radius of the circle.

Expressing the Total Length:

$400 = 2(90) + \pi \cdot d$ where $d = 2r$ (diameter of the circle).

Simplify: $400 = 180 + \pi \cdot d$

Solving for $d$:

$\pi \cdot d = 400 - 180$ $\pi \cdot d = 220$ $d = \frac{220}{\pi}$

Using $\pi \approx 3.1416$: $d = \frac{220}{3.1416} \approx 70.0 \, \text{m}$

Final Answer:

The diameter of the semicircular sections is approximately 70.0 meters.

Would you like further explanation or visual clarification? Here are some questions to deepen your understanding:

1. Why is the track's circular section divided into two semicircles?
2. How would the solution change if the total length of the track were different?
3. What role does $\pi$ play in determining the diameter?
4. Could this problem be solved graphically? How?
5. How can this concept apply to other shapes or tracks with different designs?

Tip: Always verify your work by plugging the final result back into the original equation to ensure consistency.