Question content area top
Part 1
Two brothers leave from the same point driving in opposite​ directions, the first brother driving at 65 miles per hour and the second at 55 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 300​-mile ​range, how long will they be able to talk to each​ other?


To solve this problem, let's break it down into steps:

### Step 1: Determine the distance each brother travels.
- The first brother is driving at 65 miles per hour (mph) and has a one-hour head start. Therefore, after one hour, he has traveled:
  \[
  \text{Distance} = 65 \text{ miles}
  \]

- After this first hour, the second brother starts driving at 55 mph. From this point, both brothers are moving in opposite directions.

### Step 2: Establish the rate at which they are separating from each other.
- Since they are moving in opposite directions, their relative speed is the sum of their individual speeds:
  \[
  \text{Relative speed} = 65 \text{ mph} + 55 \text{ mph} = 120 \text{ mph}
  \]

### Step 3: Calculate the initial distance between them.
- When the second brother starts driving, the first brother is already 65 miles away.

### Step 4: Determine how long it takes for the distance between them to reach 300 miles.
- They can communicate until the distance between them reaches 300 miles. We can set up the following equation to find the time \( t \) after the second brother starts driving:
  \[
  65 + 120t = 300
  \]
- Solving for \( t \):
  \[
  120t = 300 - 65 = 235
  \]
  \[
  t = \frac{235}{120} \approx 1.96 \text{ hours}
  \]

### Step 5: Include the head start time to find the total time.
- The first brother drove alone for 1 hour, and then they drove for about 1.96 hours while they could still talk. So, the total time during which they could talk is:
  \[
  1 + 1.96 \approx 2.96 \text{ hours}
  \]

### Final Answer:
The brothers will be able to talk to each other for approximately **2.96 hours** before they move out of the 300-mile communication range.

Would you like more details on any of these steps, or have any questions?

---

**Related Questions:**
1. What would happen if the first brother had a two-hour head start?
2. How would the answer change if the communication range were increased to 400 miles?
3. What if the brothers drove at different speeds, say 60 mph and 70 mph?
4. How far apart would they be after 5 hours if they kept driving?
5. How would the result change if the first brother drove at 60 mph instead of 65 mph?

**Tip:** When dealing with problems involving relative motion, adding or subtracting the speeds of the moving objects helps determine how quickly they separate or approach each other.

Learn how to calculate the communication time between two brothers driving in opposite directions with different speeds and a head start. This problem involves relative motion, speed, distance, and time concepts. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation