A motorboat maintained a constant speed of 13 miles per hour relative to the water in going 22 miles upstream and then returning. The total time for trip was 6.5 hours. Use this I for to find the speed of the current.

We are given the problem of a motorboat traveling upstream and downstream with the total trip time provided, and we are asked to determine the speed of the current. To solve this, we'll use the concept of relative speed in water and set up an equation based on the total travel time.

### Definitions and variables:
- Let \( c \) represent the speed of the current (in miles per hour).
- The speed of the boat relative to the water is 13 miles per hour.
- The distance upstream and downstream is 22 miles each way.
- The total time for the trip is 6.5 hours.

### Concept:
- When the boat is traveling **upstream**, the current works against the boat, so the effective speed of the boat is \( 13 - c \) miles per hour.
- When the boat is traveling **downstream**, the current helps the boat, so the effective speed of the boat is \( 13 + c \) miles per hour.

### Time formula:
The time taken to travel a certain distance is given by:
\[
t = \frac{d}{v}
\]
where \( t \) is time, \( d \) is distance, and \( v \) is velocity (speed).

#### Upstream travel:
The time taken to travel 22 miles upstream at speed \( 13 - c \) is:
\[
t_{\text{up}} = \frac{22}{13 - c}
\]

#### Downstream travel:
The time taken to travel 22 miles downstream at speed \( 13 + c \) is:
\[
t_{\text{down}} = \frac{22}{13 + c}
\]

### Total time:
We are told the total time for the trip is 6.5 hours, so:
\[
t_{\text{up}} + t_{\text{down}} = 6.5
\]

Substituting the expressions for \( t_{\text{up}} \) and \( t_{\text{down}} \):
\[
\frac{22}{13 - c} + \frac{22}{13 + c} = 6.5
\]

### Solving the equation:
Now, we solve this equation for \( c \).

We will multiply through by the common denominator \( (13 - c)(13 + c) \), then simplify the equation and solve for \( c \). Let me do the math.The solutions to the equation are \( c = -9 \) and \( c = 9 \). Since the speed of the current cannot be negative, the valid solution is:

\[
c = 9 \text{ miles per hour}
\]

Thus, the speed of the current is 9 miles per hour.

Would you like further details on the steps or have any other questions?

Here are some related questions:
1. How would the total time change if the speed of the current increased?
2. What would the downstream time be if the current speed was only 5 miles per hour?
3. How would the problem change if the distance upstream was 30 miles?
4. What is the total distance covered during the trip?
5. How can you express the total time as a function of the current speed?

**Tip:** Always check if a solution makes physical sense, especially when you get multiple answers in a problem.

A motorboat maintained a constant speed of 13 miles per hour relative to the water in going 22 miles upstream and then returning. The total time for the trip was 6.5 hours. Use this information to find the speed of the current.

Solve the problem of determining the speed of the current for a motorboat traveling 22 miles upstream and downstream with a total trip time of 6.5 hours. Learn the step-by-step solution using algebraic methods, relative speed concepts, and rational equations. Suitable for students in Grades 9-12.

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