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.