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, we will break it down into steps.

### Step 1: Understand the problem
The man beats a drum at two different rates (40 beats per minute and 60 beats per minute) and moves closer to the mountain. We need to find the distance between the mountain and his initial position, as well as the velocity of sound.

### Step 2: Define the variables
- Let \( S \) be the initial distance between the man and the mountain (in meters).
- Let \( v \) be the velocity of sound (in meters per second).
- The man beats the drum at a rate of 40 beats per minute initially and then at 60 beats per minute after moving 90 meters closer to the mountain.

### Step 3: Convert the drumming rates to seconds
- The time period for 40 beats per minute:
\[ T_1 = \frac{60 \text{ seconds}}{40 \text{ beats}} = 1.5 \text{ seconds} \]
- The time period for 60 beats per minute:
\[ T_2 = \frac{60 \text{ seconds}}{60 \text{ beats}} = 1 \text{ second} \]

### Step 4: Set up the equations based on the echo
When the man is at distance \( S \):
- The time taken for the sound to travel to the mountain and back is equal to the time period of the drum beat:
\[ \frac{2S}{v} = T_1 = 1.5 \text{ seconds} \]
Rearranging gives us:
\[ 2S = 1.5v \quad \text{(Equation 1)} \]

When the man moves 90 meters closer, his new distance to the mountain is \( S - 90 \):
- The time taken for the sound to travel to the mountain and back is equal to the time period of the drum beat:
\[ \frac{2(S - 90)}{v} = T_2 = 1 \text{ second} \]
Rearranging gives us:
\[ 2(S - 90) = v \quad \text{(Equation 2)} \]

### Step 5: Solve the equations
From Equation 1:
\[ S = \frac{1.5v}{2} = 0.75v \]

From Equation 2:
\[ S - 90 = \frac{v}{2} \]
Substituting \( S \) from Equation 1 into Equation 2:
\[ 0.75v - 90 = \frac{v}{2} \]
Multiplying through by 2 to eliminate the fraction:
\[ 1.5v - 180 = v \]
Rearranging gives:
\[ 1.5v - v = 180 \]
\[ 0.5v = 180 \]
\[ v = 360 \text{ m/s} \]

Now substituting \( v \) back into the equation for \( S \):
\[ S = 0.75 \times 360 = 270 \text{ meters} \]

### 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 m/s**.