Two brothers leave from the same point driving an opposite direction the first brother driving at 70 mph and the second at 65 mph. The first brother has one hour Headstart if each brother has a communication device that allows him to talk to the other for up to a 200 mile range how long will they be able to talk to each other?

We can solve this problem using the concept of relative speed and distance.

### Step-by-step approach:

1. **Relative Speed**:
   - Since the brothers are driving in opposite directions, their relative speed is the sum of their individual speeds.
   - Relative speed = 70 mph (first brother) + 65 mph (second brother) = 135 mph.

2. **Head Start**:
   - The first brother has a 1-hour head start, driving at 70 mph.
   - In that 1 hour, the first brother would cover:
     \[
     \text{Distance covered by the first brother} = 70 \text{ miles}.
     \]

3. **Distance when communication starts**:
   - The brothers will be able to talk as long as the distance between them is within 200 miles.
   - Initially, after the 1-hour head start, the first brother is 70 miles ahead of the second.
   - To calculate how long they can still communicate, we need to determine when their distance exceeds 200 miles.

4. **Distance for communication**:
   - They will be able to talk as long as the total distance between them is \( 70 + 135t \leq 200 \), where \( t \) is the time in hours after the second brother starts driving.

5. **Solving for \( t \)**:
   - Set up the inequality:
     \[
     70 + 135t = 200
     \]
   - Subtract 70 from both sides:
     \[
     135t = 130
     \]
   - Solve for \( t \):
     \[
     t = \frac{130}{135} \approx 0.96 \text{ hours}.
     \]
   - Converting this to minutes:
     \[
     0.96 \times 60 \approx 57.6 \text{ minutes}.
     \]

### Conclusion:
The brothers will be able to communicate for approximately **57.6 minutes** after the second brother starts driving.

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Would you like more details on any step, or do you have any questions?

Here are five related questions to expand on this concept:
1. What is the total distance covered by each brother during the communication time?
2. How would the solution change if the communication range was 300 miles?
3. If both brothers had started at the same time, how long would they communicate within a 200-mile range?
4. How does the relative speed change if the brothers were driving in the same direction?
5. What if the second brother drove faster than the first?

**Tip**: When solving motion problems, always account for head starts by calculating initial distances separately before considering relative speed.

Two brothers leave from the same point driving in opposite directions. The first brother is driving at 70 mph and the second at 65 mph. The first brother has a one-hour head start. If each brother has a communication device that allows them to talk for up to a 200-mile range, how long will they be able to talk to each other?

Solve a real-world relative speed problem where two brothers drive in opposite directions. With one hour head start and communication devices that have a 200-mile range, learn how to calculate the duration for which they can communicate. This problem is suitable for students in Grades 9-10, covering concepts such as relative speed and linear equations.

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