Injured runners train on a special track at a rehab center the track is a square with a half circle ont he left and right side a lap is the distance around the track the eara of the square is 138 square feet about how long is one lap use the table to help you answer the questions.
square-11.0^2=121.0,11.1^2=123.2,11.2^2=125.4,11.3^2=127.7,11.4^2=130.0,11.5^2=132.3         Square-11.6^2=134.6,11.7^2=136.9,11.8^2=139.2,11.9^2=141.6,12.0^2=144.0         answer these questions too The length of one lap around the track includes the length of

sides of the square and the distance around

circle(s).


The length of a square’s side is the square root of its area. According to the table, the side length of the square is about

feet.


The total length of the square’s sides included in the length of one lap is about

feet.


The circumference of (distance around) a circle is π times the diameter. A side of the square is the diameter of each half circle. The circumference of the circle is about
feet.

The total length of the two half circles included in one lap is about
feet.

A reasonable estimate for the length of one lap around the track, including the straight and circular portions, is

feet.

### Step-by-Step Breakdown:

We are given a track that consists of a square with half circles on the left and right sides. The area of the square is provided as **138 square feet**, and we need to estimate the total length of one lap around the track.

#### 1. **Determine the side length of the square:**
The side length of the square is the square root of its area. From the table provided, we need to find a side length that corresponds to an area of **138 square feet**. The values in the table give:

\[
11.7^2 = 136.9 \quad \text{and} \quad 11.8^2 = 139.2
\]

Since **138 square feet** lies between **11.7 feet** and **11.8 feet**, we estimate the side length to be approximately **11.75 feet**.

#### 2. **Find the total length of the square’s sides included in one lap:**
Since the square has four sides, but only the top and bottom sides are included in the lap (as the left and right sides are replaced by half circles), the total length of the square's sides in one lap is:

\[
2 \times 11.75 = 23.5 \, \text{feet}
\]

#### 3. **Determine the circumference of the half circles:**
Each half circle has a diameter equal to the side length of the square, which is **11.75 feet**. The circumference \(C\) of a full circle is given by:

\[
C = \pi \times \text{diameter}
\]

Thus, the circumference of one full circle is:

\[
C = \pi \times 11.75 \approx 3.14 \times 11.75 = 36.905 \, \text{feet}
\]

Since the lap includes **two half circles**, the total length of the curved sections is equal to the circumference of one full circle, which is approximately **36.9 feet**.

#### 4. **Estimate the total length of one lap:**
The total length of one lap around the track is the sum of the straight sides and the curved sections. Therefore, the total lap length is:

\[
\text{Lap length} = 23.5 \, \text{feet} \, (\text{straight sides}) + 36.9 \, \text{feet} \, (\text{curved sections}) = 60.4 \, \text{feet}
\]

### Final Answer:

A reasonable estimate for the length of one lap around the track is about **60.4 feet**.

---

### Questions to Further Explore:

1. How does changing the side length of the square affect the total lap length?
2. What would happen to the lap length if the circles were full instead of half?
3. How would the calculation change if the square area was slightly smaller or larger, say 140 square feet?
4. What is the percentage of the lap length that is made up by the circular sections compared to the straight sections?
5. If the area of the square doubled, how would that impact the total lap length?

### Tip:
When estimating areas or lengths involving circles, always remember to use \( \pi \approx 3.14 \) for quick calculations unless more precision is needed.

Injured runners train on a special track at a rehab center. The track is a square with a half-circle on the left and right sides. A lap is the distance around the track. The area of the square is 138 square feet. About how long is one lap? Use the table to help you answer the questions. The length of a square’s side is the square root of its area. According to the table, the side length of the square is about feet. The total length of the square’s sides included in the length of one lap is about feet. The circumference of (distance around) a circle is π times the diameter. A side of the square is the diameter of each half circle. The circumference of the circle is about feet. The total length of the two half circles included in one lap is about feet. A reasonable estimate for the length of one lap around the track, including the straight and circular portions, is feet.

Estimate the lap length of a track consisting of a square with half circles on its sides. Given an area of 138 square feet, learn how to calculate the side length of the square and use the circumference formula to find the total lap distance. Perfect for middle school students exploring geometry concepts.

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