alex runs 360 meters. The first half of the time, his speed is 5 m/s. During the second half of the time, his speed is 4 m/s. How many seconds does it take for alex to run the second half of the circumference?

To solve this problem, we need to find out how long it takes Alex to run the second half of the total time.

### Step 1: Total Time Splitting

Since Alex's run is divided into two equal halves in terms of **time**, the first half of the time is at 5 m/s and the second half is at 4 m/s. Let's denote the total time it takes to complete the 360 meters as \( T \).

### Step 2: Distance Covered in Each Half

During the first half of the total time (\( \frac{T}{2} \)), Alex is running at a speed of 5 m/s, and during the second half of the total time (\( \frac{T}{2} \)), Alex is running at a speed of 4 m/s. The total distance covered is 360 meters, so we need to calculate how much distance Alex runs during each half.

1. **First half:**
   - Time: \( \frac{T}{2} \)
   - Speed: 5 m/s
   - Distance: \( 5 \times \frac{T}{2} = \frac{5T}{2} \)

2. **Second half:**
   - Time: \( \frac{T}{2} \)
   - Speed: 4 m/s
   - Distance: \( 4 \times \frac{T}{2} = \frac{4T}{2} \)

The total distance covered is the sum of these two parts:
\[
\frac{5T}{2} + \frac{4T}{2} = 360
\]
Simplifying:
\[
\frac{9T}{2} = 360
\]
Multiply both sides by 2:
\[
9T = 720
\]
Now, divide by 9 to find \( T \):
\[
T = 80 \, \text{seconds}
\]

### Step 3: Time for the Second Half

The second half of the run takes \( \frac{T}{2} \) seconds. So, the time for the second half is:
\[
\frac{T}{2} = \frac{80}{2} = 40 \, \text{seconds}
\]

Thus, it takes Alex **40 seconds** to run the second half of the time.

---

Would you like further details or have any questions? 

Here are 5 related questions to explore:

1. How much distance does Alex cover in the first 40 seconds?
2. What is Alex's average speed for the entire 360-meter run?
3. How would the total time change if Alex's speed in the second half were increased to 6 m/s?
4. If Alex maintained a constant speed of 4.5 m/s throughout, how long would it take him to complete the 360 meters?
5. How does the time change if the total distance were 500 meters but the speeds remain the same?

**Tip:** When dealing with motion problems, carefully distinguish between time and distance when splitting different phases of the journey.

Alex runs 360 meters. The first half of the time, his speed is 5 m/s. During the second half of the time, his speed is 4 m/s. How many seconds does it take for Alex to run the second half of the circumference?

Solve how long it takes Alex to complete the second half of a 360-meter run, given varying speeds of 5 m/s and 4 m/s in the first and second halves. The problem explores algebraic concepts related to speed, time, and distance, suitable for students in Grades 6-8.

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