A man standing in front of a mountain beats a drum at regular intervals. The drumming rate is gradually increased and he finds that echo is not heard distinctly when the rate becomes `40` per minute. He then moves near to the mountain by `90` metres and finds that echo is again not heard distinctly when the drumming rate becomes `60` per minute. Calculate (a) the distance between the mountain and the initial position of the man and (b) the velocity of sound.

Explanation

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Define Variables
Let:
- \( x \) = distance from the man to the mountain (in meters)
- \( v \) = velocity of sound (in meters per second)

### Step 2: Analyze the First Scenario
When the man beats the drum at a rate of 40 beats per minute:
- The interval between beats \( T_1 \) is given by:
\[ T_1 = \frac{60 \text{ seconds}}{40 \text{ beats}} = 1.5 \text{ seconds} \]
- The time taken for the echo to return is:
\[ t_1 = \frac{2x}{v} \]
- Since the echo is not heard distinctly, the time for the echo to return must equal the interval between beats:
\[ \frac{2x}{v} = 1.5 \]
- Rearranging gives us our first equation:
\[ 2x = 1.5v \quad \text{(Equation 1)} \]

### Step 3: Analyze the Second Scenario
After moving 90 meters closer to the mountain, the man beats the drum at a rate of 60 beats per minute:
- The interval between beats \( T_2 \) is given by:
\[ T_2 = \frac{60 \text{ seconds}}{60 \text{ beats}} = 1 \text{ second} \]
- The new distance to the mountain is \( x - 90 \) meters, so the time taken for the echo to return is:
\[ t_2 = \frac{2(x - 90)}{v} \]
- Setting this equal to the new interval gives us:
\[ \frac{2(x - 90)}{v} = 1 \]
- Rearranging gives us our second equation:
\[ 2(x - 90) = v \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( 2x = 1.5v \)
2. \( 2(x - 90) = v \)

From Equation 2:
\[ 2x - 180 = v \quad \text{(Rearranging)} \]
Now substitute \( v \) from Equation 2 into Equation 1:
\[ 2x = 1.5(2x - 180) \]
Expanding this gives:
\[ 2x = 3x - 270 \]
Rearranging leads to:
\[ x = 270 \text{ meters} \]

### Step 5: Find the Velocity of Sound
Now substitute \( x \) back into Equation 2 to find \( v \):
\[ v = 2(270 - 90) = 2(180) = 360 \text{ meters/second} \]

### Final Answers
(a) The distance between the mountain and the initial position of the man is **270 meters**.
(b) The velocity of sound is **360 meters/second**.