A motorboat maintained a constant speed of 10 miles per hour relative to the water in going 9 miles upstream and then returning. The total time for the trip was 5.0 hours. Use this information to find the speed of the current.

To solve for the speed of the current, we can set up the problem using the relationship between distance, speed, and time. Let's break it down:

### Variables:
- Let \( v_c \) be the speed of the current in miles per hour.
- The speed of the boat relative to the water is 10 miles per hour.
- The distance upstream (and downstream) is 9 miles.

### Time for Upstream Travel:
When going upstream, the boat's effective speed is reduced by the speed of the current:
\[
\text{Speed upstream} = 10 - v_c
\]
\[
\text{Time upstream} = \frac{\text{Distance upstream}}{\text{Speed upstream}} = \frac{9}{10 - v_c}
\]

### Time for Downstream Travel:
When going downstream, the boat's effective speed is increased by the speed of the current:
\[
\text{Speed downstream} = 10 + v_c
\]
\[
\text{Time downstream} = \frac{\text{Distance downstream}}{\text{Speed downstream}} = \frac{9}{10 + v_c}
\]

### Total Time for the Trip:
The total time for the trip is given as 5 hours. Therefore, we have:
\[
\frac{9}{10 - v_c} + \frac{9}{10 + v_c} = 5
\]

### Solving the Equation:
To solve this equation for \( v_c \), follow these steps:

1. Multiply through by the common denominator \( (10 - v_c)(10 + v_c) \):
\[
9(10 + v_c) + 9(10 - v_c) = 5(10 - v_c)(10 + v_c)
\]

2. Simplify:
\[
90 + 9v_c + 90 - 9v_c = 5(100 - v_c^2)
\]
\[
180 = 500 - 5v_c^2
\]

3. Solve for \( v_c^2 \):
\[
5v_c^2 = 500 - 180 = 320
\]
\[
v_c^2 = \frac{320}{5} = 64
\]
\[
v_c = \sqrt{64} = 8
\]

### Final Answer:
The speed of the current \( v_c \) is **8 miles per hour**.

Would you like more details or have any questions? Here are some related questions to explore further:

1. What would the total time be if the current speed was 5 mph instead of 8 mph?
2. How would the problem change if the boat's speed relative to the water was 15 mph?
3. What if the boat only traveled upstream? How would we calculate the time then?
4. Can you find the current speed if the total distance traveled was different?
5. How would the problem be affected if the total time was given as 6 hours instead of 5?

**Tip:** When solving problems involving relative speed, carefully consider the direction of movement (upstream or downstream) to correctly apply the current's effect on the overall speed.

Discover how to find the speed of the current when a motorboat travels 9 miles upstream and returns in 5 hours at a constant speed of 10 miles per hour relative to the water. Follow a step-by-step solution involving relative speed equations and calculate the current's speed effectively.

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