There are two boats B1 and B2, where actual speed of B1 is 66.66% more than that of B2. Speed of stream is K km/h.

When both boats started from point P1 (downstream), Boat B2 started M/2 hours earlier than B1. After (M + 7.5) hours when B2 started, distance covered by B1 is more than B2.

When B1 started from point P2 and B2 started from point P1, in opposite direction, at same time they both meets first time after (2K + M – 6) hours

Note:

a) 2K2 + 38K – 432 = 0

b) 18, 54, 108, (?), 270, 378

M = (remainder obtained when missing number in the series is divided by 13 – 6)

Both boats are rowing in same river.When B1 started from point P1, and B2 started from point P2, then find the total time for their second meeting?

To solve this problem, we need to break it down into parts:

### Step 1: Solve for \( K \)
Given the quadratic equation:
\[
2K^2 + 38K - 432 = 0
\]
We need to solve this quadratic equation for \( K \).

### Step 2: Determine the missing number in the series and calculate \( M \)
The series is: 18, 54, 108, (?), 270, 378.

The pattern in the series suggests multiplying by a certain factor. The pattern is:
- \( 18 \times 3 = 54 \)
- \( 54 \times 2 = 108 \)
- \( 108 \times 2.5 = 270 \) 
- \( 270 \times 1.4 \approx 378 \)

The missing number is \( 108 \times 2 = 216 \).

Now, find \( M \):
\[
M = \left( \text{remainder of } \frac{216}{13} \right) - 6
\]
Calculate the remainder and determine \( M \).

### Step 3: Determine the relationship between the boats' speeds
Given that the speed of B1 is 66.66% more than B2:
\[
\text{Let speed of B2 be } v \text{, then speed of B1 is } \frac{5}{3}v.
\]

### Step 4: Use the given conditions to establish equations and solve
From the condition of them meeting after a certain time when starting from P2 and P1 respectively, we need to use the time and speed information to solve for the required meeting times.

### Step 5: Calculate the total time for their second meeting.

Let's begin with solving the quadratic equation for \( K \) and then proceed with each step as outlined. I'll calculate this and provide the final solution.The solutions for \( K \) from the quadratic equation are \( K = -27 \) and \( K = 8 \). Since speed cannot be negative, we select \( K = 8 \).

### Step 2: Find \( M \) using the series
The missing number in the series was found to be 216. Let's calculate \( M \) now:
\[
M = \left( \text{remainder of } \frac{216}{13} \right) - 6
\]
I'll compute this next.The value of \( M \) is 2.

### Step 3: Analyzing the Problem
With \( K = 8 \) km/h and \( M = 2 \) hours, we can analyze the meeting conditions of the boats.

### Step 4 & 5: Determine the total time for their second meeting
We know:
- Speed of B2 is \( v \), and speed of B1 is \( \frac{5}{3}v \).
- They meet after \( 2K + M - 6 \) hours when starting in opposite directions.

Now, I will compute the time taken for their second meeting when they start from opposite directions.Given the values:
- \( K = 8 \) km/h
- \( M = 2 \) hours
- Speed of B1 is \( \frac{5}{3}v \)
- Speed of B2 is \( v \)

They first meet after \( 2K + M - 6 \) hours:
\[
\text{Time for first meeting} = 2(8) + 2 - 6 = 16 + 2 - 6 = 12 \text{ hours}
\]

For the second meeting in the same river:
- B1 and B2 are now moving in opposite directions.
- They will meet again after the same time interval if they continue rowing.

Thus, the total time for the second meeting will be double the time for the first meeting.

\[
\text{Total time for second meeting} = 12 \times 2 = 24 \text{ hours}
\]

### Conclusion:
The total time for the second meeting of the two boats is 24 hours.

Would you like more details or have any further questions?

---

### Additional Practice Questions:
1. What is the speed of B2 if the actual speed of B1 is 20 km/h?
2. How would the time change if the speed of the stream \( K \) was doubled?
3. What is the relationship between the stream speed \( K \) and the time taken for the first meeting?
4. If B2 started earlier by 1 hour instead of \( M/2 \) hours, how would the situation change?
5. How would you set up the problem if the boats were traveling upstream instead of downstream?
6. Calculate the time taken for the boats to meet if \( K = 6 \) km/h and \( M = 3 \).
7. What would happen if the speed of B1 was only 50% more than B2?
8. How would the time for the second meeting change if B1 started later?

### Tip:
When dealing with relative speed problems, always clearly define the direction of motion and speed relative to the current or stream. This helps in setting up the equations correctly.

Learn how to calculate the total time for the second meeting of boats B1 and B2 rowing in a river. This problem involves relative speed calculations, quadratic equations, and time-distance relationships. Suitable for high school students.

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