A hunter fired two shots from the branch of a tree at an interval of 76 seconds. A tiger separating too fast hears the two shots at an interval of 83 seconds. If the velocity of the sound is 1195.2 km/h, then find the speed of tiger?

Options

A. 112.8 km/h
B. 100.8 km/h
C. 80.16 km/h
D. none of these

Answer

To solve the problem, we need to determine the speed of the tiger based on the information given about the shots fired by the hunter and the time intervals involved. Here’s a step-by-step solution:

### Step 1: Understand the Problem
The hunter fires two shots at an interval of 76 seconds. The tiger hears these shots at an interval of 83 seconds. The speed of sound is given as 1195.2 km/h.

### Step 2: Convert the Speed of Sound to m/s
To make calculations easier, we should convert the speed of sound from km/h to m/s.
1 km/h = (1000 m / 3600 s) = 5/18 m/s.

So,
\[ \text{Speed of sound in m/s} = 1195.2 \times \frac{5}{18} \]
\[ = 1195.2 \times 0.27778 \]
\[ \approx 332.0 \text{ m/s} \]

### Step 3: Set Up the Equations
Let \( v \) be the speed of the tiger in m/s.
- The time taken for the sound of the first shot to reach the tiger is \( t_1 \).
- The time taken for the sound of the second shot to reach the tiger is \( t_2 \).

From the problem, we know:
- \( t_1 = \frac{d}{v_s} \) (where \( d \) is the distance to the tiger when the first shot is fired)
- \( t_2 = \frac{d + vt}{v_s} \) (where \( vt \) is the distance the tiger covers in the time interval of 76 seconds)

Since the shots are fired 76 seconds apart, we have:
\[ t_2 - t_1 = 83 \text{ seconds} \]

### Step 4: Express Distances
From the first shot:
\[ t_1 = \frac{d}{332} \]

From the second shot:
\[ t_2 = \frac{d + 76v}{332} \]

### Step 5: Set Up the Equation
Now we can set up the equation based on the time difference:
\[ \frac{d + 76v}{332} - \frac{d}{332} = 83 \]

### Step 6: Simplify the Equation
Multiply through by 332 to eliminate the denominator:
\[ (d + 76v) - d = 83 \times 332 \]
\[ 76v = 83 \times 332 \]
\[ 76v = 27556 \]

### Step 7: Solve for the Speed of the Tiger
Now, divide both sides by 76:
\[ v = \frac{27556}{76} \]
\[ v \approx 362.0 \text{ m/s} \]

### Step 8: Convert Back to km/h
To convert back to km/h:
\[ v = 362.0 \times \frac{18}{5} \]
\[ v \approx 1303.2 \text{ km/h} \]

### Final Answer
The speed of the tiger is approximately **1303.2 km/h**.

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