The problem involves a boat traveling up and down a river, with the current affecting its speed. Let's define the variables to solve it:

• Let the speed of the boat in still water be $5 \, \text{mph}$.
• Let the speed of the current be $c \, \text{mph}$.

Upstream trip:

• When going upstream, the boat's effective speed is reduced by the current, so its speed is $5 - c \, \text{mph}$.
• The distance traveled upstream is $15 \, \text{miles}$.

The time taken to travel upstream is: $t_1 = \frac{15}{5 - c}$

Downstream trip:

• When returning downstream, the boat's speed is increased by the current, so its speed is $5 + c \, \text{mph}$.
• The distance traveled downstream is also $15 \, \text{miles}$.

The time taken to travel downstream is: $t_2 = \frac{15}{5 + c}$

Total time:

The total time for the round trip is given as 8 hours, so: $t_1 + t_2 = 8$ Substituting the expressions for $t_1$ and $t_2$: $\frac{15}{5 - c} + \frac{15}{5 + c} = 8$

Solving for $c$:

To solve this, multiply both sides by $(5 - c)(5 + c)$ to eliminate the denominators: $15(5 + c) + 15(5 - c) = 8(5^2 - c^2)$ Simplifying: $15(5 + c) + 15(5 - c) = 8(25 - c^2)$ $75 + 15c + 75 - 15c = 200 - 8c^2$ $150 = 200 - 8c^2$ Now, solve for $c^2$: $8c^2 = 50$ $c^2 = \frac{50}{8} = 6.25$ $c = \sqrt{6.25} = 2.5$

Thus, the average speed of the current is $2.5 \, \text{mph}$.

Would you like further details on the solution or explanation of any specific step?

Here are 5 questions to expand on this:

1. How would the total time change if the current speed increased?
2. What if the boat's speed in still water was reduced by 1 mph—how would that affect the answer?
3. How do you solve similar problems where the distances upstream and downstream are different?
4. Could we approach this problem using a system of equations? How?
5. How can this method be applied to real-life problems involving transportation with resistance?

Tip: Always check for unit consistency (like miles per hour vs miles or hours) when solving word problems!