A motor boat can maintain a constant speed of 19 miles per hour relative to the water. The boat makes a trip upstream to a certain point in 55 minutes the return trip takes 40 minutes. What is the speed of the current.

Let the speed of the current be \( c \) miles per hour. The speed of the boat relative to the water is 19 miles per hour.

### Step 1: Set up the relationships for the upstream and downstream trips.

1. **Upstream:**
   When the boat is traveling upstream, it moves against the current, so its effective speed is \( 19 - c \). The time taken for the upstream trip is 55 minutes, or \( \frac{55}{60} = \frac{11}{12} \) hours.

   Let the distance to the point be \( d \) miles. So,
   \[
   d = \left( 19 - c \right) \times \frac{11}{12}
   \]

2. **Downstream:**
   When the boat is traveling downstream, it moves with the current, so its effective speed is \( 19 + c \). The time taken for the downstream trip is 40 minutes, or \( \frac{40}{60} = \frac{2}{3} \) hours.

   The distance for the downstream trip is the same \( d \) miles, so,
   \[
   d = \left( 19 + c \right) \times \frac{2}{3}
   \]

### Step 2: Set up an equation.
Since the distance \( d \) is the same for both upstream and downstream trips, we can set the two expressions for \( d \) equal to each other:
\[
\left( 19 - c \right) \times \frac{11}{12} = \left( 19 + c \right) \times \frac{2}{3}
\]

### Step 3: Solve for \( c \).
To eliminate the fractions, multiply both sides by 12:
\[
11 \times \left( 19 - c \right) = 8 \times \left( 19 + c \right)
\]

Expanding both sides:
\[
209 - 11c = 152 + 8c
\]

Move the terms involving \( c \) to one side and the constants to the other side:
\[
209 - 152 = 8c + 11c
\]
\[
57 = 19c
\]

Solve for \( c \):
\[
c = \frac{57}{19} = 3 \, \text{miles per hour}
\]

### Final Answer:
The speed of the current is \( 3 \) miles per hour.

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

Here are 5 related questions to explore further:
1. How would the problem change if the boat's speed relative to the water was 25 mph?
2. What happens if the current speed increases by 2 mph?
3. How long would the upstream trip take if the current speed was halved?
4. What would the speed of the boat need to be to complete both trips in exactly 1 hour?
5. If the boat travels in a river with no current, how does the total travel time change?

**Tip:** When dealing with problems involving relative speeds (like boats or planes in wind), always consider the combined effects of the moving medium (current or wind) on the object's speed.

A motor boat can maintain a constant speed of 19 miles per hour relative to the water. The boat makes a trip upstream to a certain point in 55 minutes, the return trip takes 40 minutes. What is the speed of the current?

Learn how to solve the boat speed vs current problem: A boat moving at 19 mph takes 55 minutes upstream and 40 minutes downstream. Find the speed of the current using algebra and relative speed concepts. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation