A race track an area of 49500 m2. The circumference of the track is 1000 m. The sides are of equal length and parallel to each other and the ends of the race track are semicircles. Find the length of the track area  x

To solve this problem, let's break down the track's geometry. The race track consists of:

1. Two straight sides of equal length.
2. Two semicircles at the ends.

### Step 1: Identify the key variables

Let:
- \( x \) be the length of each straight side.
- The total area of the track is 49,500 m².
- The total circumference (perimeter) of the track is 1,000 m.

### Step 2: Break the circumference equation

The total circumference consists of:
- Two straight sides of length \( x \).
- The circumference of two semicircles, which together form a full circle.

The circumference of a circle is \( 2\pi r \), where \( r \) is the radius of the semicircles.

Thus, the total circumference is:
\[
2x + 2\pi r = 1000
\]
Simplifying this equation:
\[
x + \pi r = 500 \quad \text{(Equation 1)}
\]

### Step 3: Break the area equation

The total area of the track consists of:
- The area of the rectangular part, which is \( 2xr \) (the length \( x \) and width \( 2r \), since the width of the track is the diameter of the semicircles).
- The area of the two semicircles, which together form a full circle. The area of a circle is \( \pi r^2 \).

Thus, the total area is:
\[
2xr + \pi r^2 = 49500 \quad \text{(Equation 2)}
\]

### Step 4: Solve the system of equations

We now have two equations:
1. \( x + \pi r = 500 \)
2. \( 2xr + \pi r^2 = 49500 \)

Let’s solve these step by step.

#### 1. Solve for \( x \) in terms of \( r \) from Equation 1:
\[
x = 500 - \pi r
\]

#### 2. Substitute this into Equation 2:
\[
2(500 - \pi r)r + \pi r^2 = 49500
\]
Expanding this:
\[
1000r - 2\pi r^2 + \pi r^2 = 49500
\]
Simplify:
\[
1000r - \pi r^2 = 49500
\]

### Step 5: Solve for \( r \)

We now have a quadratic equation in terms of \( r \):
\[
\pi r^2 - 1000r + 49500 = 0
\]
Use the quadratic formula to solve for \( r \):
\[
r = \frac{-(-1000) \pm \sqrt{(-1000)^2 - 4(\pi)(49500)}}{2(\pi)}
\]

Let’s calculate this next.The two possible values for the radius \( r \) are approximately:
- \( r_1 = 257.00 \, \text{m} \)
- \( r_2 = 61.31 \, \text{m} \)

Since the radius of the semicircles should be smaller than half of the total circumference divided by \( \pi \), the valid solution is \( r = 61.31 \, \text{m} \).

### Step 6: Calculate \( x \)

Now, using Equation 1 \( x + \pi r = 500 \), substitute \( r = 61.31 \):
\[
x = 500 - \pi(61.31)
\]
Let’s calculate \( x \).The length of each straight side \( x \) is approximately \( 307.39 \, \text{m} \).

### Summary:
- The radius of the semicircles is \( 61.31 \, \text{m} \).
- The length of each straight side is \( 307.39 \, \text{m} \).

Do you have any questions or would you like more details on any part of the solution?

Here are 5 related questions to explore further:
1. How would the formula change if the track had elliptical ends instead of semicircles?
2. What would be the total length of the straight sections if the track's area was increased by 10%?
3. How would you calculate the maximum radius if the circumference remained constant but the area increased?
4. If the straight sections were twice as long, how would this affect the overall area of the track?
5. How does the area-to-circumference ratio change as the semicircle radius decreases?

**Tip:** The quadratic formula is useful when solving equations where variables are squared, such as in area or volume problems with curved shapes like circles or ellipses.

A race track has an area of 49500 m². The circumference of the track is 1000 m. The sides are of equal length and parallel to each other, and the ends of the race track are semicircles. Find the length of the track area x.

Learn how to solve a geometry problem involving a race track with an area of 49500 m² and a circumference of 1000 m. This solution covers key concepts such as semicircles, straight sides, and applying the quadratic formula. Suitable for students in Grades 9-11.

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