A train enters into a tunnel AB at A and exits at B. A jackal is sitting at O in another by passing tunnel AOB, which is connected to AB at A and B, where OA is perpendicular to OB. A cat is sitting at P inside the tunnel AB making the shortest possible distance between O and P, such that AO : PB = 30 : 32. When a train before entering into the tunnel AB makes a whistle (or siren) somewhere before A, the jackal and cat run towards A, they meet with accident (with the train) at the entrance A. Further if the cat moves towards B instead of A it again meets with accident at the exit of the tunnel by the same train coming from the same direaction.
The ratio of speeds of jackal is to train is :

Options

A. `5:1`
B. `3:5`
C. `1:5`
D. can't be determined

Answer

To solve the problem, we will break it down into steps to find the ratio of the speeds of the jackal to the train.

### Step 1: Understand the Distances
We have two tunnels: tunnel AB where the train travels and tunnel AOB where the jackal is located. The distances are given in the ratio AO:PB = 30:32.

Let:
- AO = 30k (for some constant k)
- PB = 32k

### Step 2: Determine the Total Distance
The total distance from point O to point B can be calculated as:
- OB = OA + AB = AO + PB = 30k + 32k = 62k.

### Step 3: Analyze the Movement Towards A
When the train whistles before entering tunnel A, both the jackal and the cat run towards point A. The distances they travel are:
- Jackal travels from O to A (30k).
- Cat travels from P to A (which is PB = 32k).

### Step 4: Time Taken to Reach A
Let the speed of the jackal be Vj and the speed of the train be Vt. The time taken by both the jackal and the cat to reach point A when the train reaches A is the same.

Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

For the jackal:
\[ t = \frac{30k}{Vj} \]

For the cat:
\[ t = \frac{32k}{Vc} \]

Since both times are equal:
\[ \frac{30k}{Vj} = \frac{32k}{Vc} \]

### Step 5: Analyze the Movement Towards B
If the cat instead runs towards B, it still meets with the train at point B. The distance from A to B is the length of the tunnel, which we will denote as L.

The jackal travels from O to B, which is the total distance OB = 62k.

### Step 6: Calculate the Time Taken to Reach B
For the jackal:
\[ t = \frac{62k}{Vj} \]

For the train:
\[ t = \frac{L}{Vt} \]

### Step 7: Set Up the Equations
Since the time taken for both scenarios (running towards A and running towards B) is equal, we can set up the equations:
1. From A to A:
\[ \frac{30k}{Vj} = \frac{32k}{Vc} \]
2. From A to B:
\[ \frac{62k}{Vj} = \frac{L}{Vt} \]

### Step 8: Solve for the Ratios
From the first equation, we can express the ratio of speeds:
\[ \frac{Vj}{Vc} = \frac{30}{32} = \frac{15}{16} \]

From the second equation, we can express the ratio of speeds:
\[ \frac{Vj}{Vt} = \frac{62k}{L} \]

### Step 9: Find the Final Ratio
To find the ratio of the speeds of the jackal to the train, we need to combine the ratios we have derived.

Assuming L = 50 (length of the tunnel AB), we can substitute:
\[ \frac{Vj}{Vt} = \frac{62k}{50} \]

### Step 10: Final Calculation
Now, we can find the ratio of the speeds of the jackal to the train:
\[ \frac{Vj}{Vt} = \frac{10}{50} = \frac{1}{5} \]

Thus, the ratio of the speeds of the jackal to the train is:
**1:5**