Two brothers leave from the same point driving in opposite​ directions, the first brother driving at 65 miles per hour and the second at 60 mph. The first brother has a​ one-hour head start. If each brother has a communication device that allows him to talk to the other for up to a 220​-mile ​range, how long will they be able to talk to each​ other?

Let's break down the problem step by step.

### Step 1: Define Variables
- Speed of the first brother: \( 65 \, \text{mph} \)
- Speed of the second brother: \( 60 \, \text{mph} \)
- Head start time of the first brother: 1 hour
- Range of communication: 220 miles
- Let \( t \) represent the time after the second brother starts driving that they can still communicate.

### Step 2: Distance Covered by Each Brother
- In the first hour, the first brother covers \( 65 \, \text{miles} \) (since he drives for one hour at 65 mph).
- After the second brother starts, both brothers are moving apart from each other, with their relative speed being the sum of their individual speeds: \( 65 + 60 = 125 \, \text{mph} \).

### Step 3: Distance Formula
The two brothers will be able to talk to each other until the total distance between them reaches 220 miles. So, we need to find the time \( t \) when the total distance between them (after the second brother starts driving) reaches 220 miles.

- Initial distance between them due to the head start: 65 miles.
- After that, they separate at \( 125 \, \text{mph} \).
  
The equation for the total distance between them after \( t \) hours is:
\[
\text{Total Distance} = 65 + 125t
\]
Set the total distance equal to 220 miles (since that is the communication range):
\[
65 + 125t = 220
\]

### Step 4: Solve for \( t \)
\[
125t = 220 - 65
\]
\[
125t = 155
\]
\[
t = \frac{155}{125} = 1.24 \, \text{hours}
\]

### Conclusion:
The brothers will be able to talk to each other for approximately **1.24 hours** after the second brother starts driving.

### Do you need further details or clarification on any part of this problem?

Here are 5 related questions to consider:
1. How would the answer change if the communication range was 300 miles?
2. What if the first brother drove at 70 mph instead of 65 mph?
3. How long would they be able to talk if the second brother started driving at 75 mph?
4. What is the total distance each brother covers by the time they are out of range?
5. How would this situation differ if they both drove in the same direction?

**Tip:** In problems involving relative motion, always calculate the combined rate at which the two objects are separating or closing the distance.

Two brothers leave from the same point driving in opposite directions, the first brother driving at 65 miles per hour and the second at 60 mph. The first brother has a one-hour head start. If each brother has a communication device that allows him to talk to the other for up to a 220-mile range, how long will they be able to talk to each other?

Explore how to solve a relative motion problem where two brothers drive in opposite directions, with one brother having a one-hour head start. Learn key algebraic concepts and linear equations to find how long they can communicate within a 220-mile range. Suitable for Grades 8-10.

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