Two boats A and B are moving along perpendicular paths in a still lake at night. Boat A move with a speed of `3 m//s` and boat B moves with a speed of `4 m//s` in the direction such that they collide after sometime. At `t=0`, the boats are `300 m` apart. The ratio of distance of distance travelled by boat A to the distance travelled by boat B at the instant of collision is :-

Options

A. 1
B. `1//2`
C. `3//4`
D. `4//3`

Explanation

To solve the problem step by step, we will analyze the motion of the two boats and find the required ratio of distances traveled by each boat at the moment of collision.

### Step 1: Understand the Problem
We have two boats, A and B, moving at right angles to each other. Boat A moves at a speed of 3 m/s, and Boat B moves at a speed of 4 m/s. Initially, they are 300 meters apart.

### Step 2: Set Up the Diagram
Draw a right triangle where:
- One leg represents the distance traveled by Boat A.
- The other leg represents the distance traveled by Boat B.
- The hypotenuse represents the initial distance between the two boats (300 m).

### Step 3: Use the Pythagorean Theorem
According to the Pythagorean theorem, the relationship between the sides of the triangle can be expressed as:
\[ d_A^2 + d_B^2 = 300^2 \]
where \(d_A\) is the distance traveled by Boat A and \(d_B\) is the distance traveled by Boat B.

### Step 4: Express Distances in Terms of Time
The distances traveled by the boats can be expressed as:
- For Boat A: \(d_A = 3t\)
- For Boat B: \(d_B = 4t\)

### Step 5: Substitute Distances in the Pythagorean Theorem
Substituting \(d_A\) and \(d_B\) into the Pythagorean theorem gives:
\[ (3t)^2 + (4t)^2 = 300^2 \]
This simplifies to:
\[ 9t^2 + 16t^2 = 90000 \]
\[ 25t^2 = 90000 \]

### Step 6: Solve for Time \(t\)
Now, divide both sides by 25:
\[ t^2 = \frac{90000}{25} = 3600 \]
Taking the square root of both sides:
\[ t = 60 \text{ seconds} \]

### Step 7: Calculate Distances Traveled
Now we can find the distances traveled by each boat:
- Distance traveled by Boat A:
\[ d_A = 3t = 3 \times 60 = 180 \text{ meters} \]
- Distance traveled by Boat B:
\[ d_B = 4t = 4 \times 60 = 240 \text{ meters} \]

### Step 8: Find the Ratio of Distances
The ratio of the distance traveled by Boat A to the distance traveled by Boat B is:
\[ \text{Ratio} = \frac{d_A}{d_B} = \frac{180}{240} \]
This simplifies to:
\[ \frac{180 \div 60}{240 \div 60} = \frac{3}{4} \]

### Final Answer
The ratio of the distance traveled by Boat A to the distance traveled by Boat B at the instant of collision is \(3:4\).

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