A motorboat, that has speed of 10 km/hour in still water, left a pier traveling against the current of the river. Forty-five minutes after the boat left the pier, the motor of the boat broke, and the boat began drifting with the current. After three hours of drifting with the current, the boat was back at the pier where it had started. What is the speed of the current of the river?

To solve this problem, we need to break it down step by step, considering the speeds of both the motorboat and the river current. Let's denote the speed of the current as "c" km/h. The speed of the motorboat in still water is given as 10 km/h. When the boat is traveling against the current, its effective speed becomes (10 - c) km/h. When it starts drifting with the current, its effective speed is (10 + c) km/h.

Understanding the Journey

The motorboat travels against the current for 45 minutes before the motor fails. We need to convert this time into hours for consistency in our calculations:

• 45 minutes = 0.75 hours

Distance Traveled Against the Current

During this time, the distance the boat covers can be calculated using the formula:

Distance = Speed × Time

Thus, the distance traveled against the current is:

Distance = (10 - c) × 0.75

Drifting with the Current

After the motor fails, the boat drifts with the current for 3 hours. The distance it covers while drifting can be calculated as:

Distance = Speed × Time

So, the distance during this phase is:

Distance = (10 + c) × 3

Setting Up the Equation

Since the boat returns to the pier, the distance traveled against the current must equal the distance traveled with the current. Therefore, we can set up the following equation:

(10 - c) × 0.75 = (10 + c) × 3

Solving the Equation

Now, let's solve for "c". First, we can expand both sides:

• Left side: 0.75(10 - c) = 7.5 - 0.75c
• Right side: 3(10 + c) = 30 + 3c

Now, we have:

7.5 - 0.75c = 30 + 3c

Next, let's isolate "c". Start by adding 0.75c to both sides:

7.5 = 30 + 3c + 0.75c

7.5 = 30 + 3.75c

Now, subtract 30 from both sides:

7.5 - 30 = 3.75c

-22.5 = 3.75c

Finally, divide both sides by 3.75 to solve for "c":

c = -22.5 / 3.75

c = -6

Interpreting the Result

Since speed cannot be negative, we take the absolute value, which means the speed of the current is 6 km/h. Therefore, the current of the river flows at a speed of 6 km/h.

This problem illustrates the importance of understanding relative speeds in different conditions, and how to set up equations based on the relationships between distances and speeds. If you have any further questions or need clarification on any part of this process, feel free to ask!