A boat takes 2 hours to travel downstream a river from port A to port B, and 
3 hours to return to port A. Another boat takes a total of 6 hours to travel 
from port B to port A and return to port B. If the speeds of the boats and the 
river are constant, then the time, in hours, taken by the slower boat to travel 
from port A to port B is

Let's break this problem down step by step.

### Step 1: Define variables
Let:
- \( v_b \) be the speed of the first boat in still water.
- \( v_r \) be the speed of the river current.
- \( d \) be the distance between port A and port B.

From the problem:
- The first boat takes 2 hours to travel downstream from A to B.
- It takes 3 hours to travel upstream from B to A.

#### Downstream speed:
The downstream speed of the first boat is \( v_b + v_r \). The time taken to travel from A to B is 2 hours, so:
\[
\frac{d}{v_b + v_r} = 2 \quad \text{(1)}
\]
#### Upstream speed:
The upstream speed of the first boat is \( v_b - v_r \). The time taken to travel from B to A is 3 hours, so:
\[
\frac{d}{v_b - v_r} = 3 \quad \text{(2)}
\]

### Step 2: Express distance \( d \)
From equation (1), we get:
\[
d = 2(v_b + v_r) \quad \text{(3)}
\]
From equation (2), we get:
\[
d = 3(v_b - v_r) \quad \text{(4)}
\]

### Step 3: Set equations equal
From equations (3) and (4), we can set the expressions for \( d \) equal to each other:
\[
2(v_b + v_r) = 3(v_b - v_r)
\]
Expand both sides:
\[
2v_b + 2v_r = 3v_b - 3v_r
\]
Rearrange terms:
\[
2v_b - 3v_b = -3v_r - 2v_r
\]
\[
-v_b = -5v_r
\]
This gives:
\[
v_b = 5v_r \quad \text{(5)}
\]

### Step 4: Solve for \( v_r \) and \( v_b \)
Substitute \( v_b = 5v_r \) into equation (3):
\[
d = 2(5v_r + v_r) = 2(6v_r) = 12v_r
\]

Now substitute \( d = 12v_r \) into equation (1):
\[
\frac{12v_r}{v_b + v_r} = 2
\]
Substitute \( v_b = 5v_r \):
\[
\frac{12v_r}{5v_r + v_r} = 2
\]
Simplify:
\[
\frac{12v_r}{6v_r} = 2
\]
\[
2 = 2
\]
This is correct, so our values for \( v_b \) and \( v_r \) are consistent.

### Step 5: Time for the second boat
Let \( v_b' \) be the speed of the second boat in still water.

From the problem, the second boat takes 6 hours for a round trip (from B to A and back to B). The time to travel from B to A is the same as from A to B, so let \( t' \) be the time the slower boat takes to travel from A to B.

The round-trip time is:
\[
t' + t' = 6
\]
So:
\[
2t' = 6
\]
This gives:
\[
t' = 3
\]

Thus, the time taken by the slower boat to travel from port A to port B is **3 hours**.

### Do you have any questions or would you like further details?

### Related Questions:
1. How do we determine the speed of the current using only the boat speeds and times?
2. What would happen to the travel times if the current's speed doubled?
3. How would the problem change if the boats had different engine powers for upstream and downstream travel?
4. Can the round-trip times provide insights into fuel consumption in real-world scenarios?
5. How do external factors like wind affect the travel time calculations?

**Tip**: Whenever working with problems involving relative motion, breaking them down into components (upstream/downstream or along axes) makes the analysis clearer.

A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is

This problem involves calculating the time taken by a slower boat to travel between two ports in a river, factoring in the effects of the current. Learn how to apply relative speed and linear equations to solve this real-world scenario. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation