Determine the Boat's Speed Over 50 km After an Initial 40-Minute Journey

To address this problem, let's follow these steps:

• Step 1: Calculate the speed for the first segment of the journey.
• Step 2: Use the overall average speed to find the total time required for the entire voyage.
• Step 3: Determine the time available for the second segment and calculate the required speed.

Step 1: For the first segment, the distance is $30$ km, and the time taken is $40$ minutes, which is $\frac{40}{60} = \frac{2}{3}$ hours. Thus, the speed for the first segment is:

Step 2: Since the average speed for the entire trip is $45$ km/h, let's find the total time required for both segments. The total distance is $30 + 50 = 80$ km. Thus, the total time required is:

Step 3: The first segment took $\frac{2}{3}$ hours. Hence, the time available for the second segment of $50$ km is:

Therefore, the speed required for the second segment is:

Thus, the speed of the boat over the next 50 km is $45$ km/h.