A person standing between two parallel hills fires a gun. He hears the first echo after `(3) /(2)` s, and a second echo after `(5) /(2)` s. If speed of sound is `332 m//s` , Calculate the distance between the hills. When will he hear the third echo?

Explanation

To solve the problem, we will break it down into steps.

### Step 1: Understanding the Echoes
When the person fires the gun, the sound travels to the first hill and back, which takes a certain amount of time. The time taken for the first echo is given as \( \frac{3}{2} \) seconds. The sound also travels to the second hill and back, taking \( \frac{5}{2} \) seconds.

### Step 2: Calculate the Distance to the First Hill
The total distance traveled by sound for the first echo is \( 2d_1 \) (to the first hill and back). Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
we can express this as:
\[ 2d_1 = v \cdot t_1 \]
where \( v = 332 \, \text{m/s} \) and \( t_1 = \frac{3}{2} \, \text{s} \).

Now substituting the values:
\[ 2d_1 = 332 \cdot \frac{3}{2} \]
\[ 2d_1 = 332 \cdot 1.5 = 498 \, \text{m} \]
Now, dividing both sides by 2 to find \( d_1 \):
\[ d_1 = \frac{498}{2} = 249 \, \text{m} \]

### Step 3: Calculate the Distance to the Second Hill
Similarly, for the second echo, we have:
\[ 2d_2 = v \cdot t_2 \]
where \( t_2 = \frac{5}{2} \, \text{s} \).

Substituting the values:
\[ 2d_2 = 332 \cdot \frac{5}{2} \]
\[ 2d_2 = 332 \cdot 2.5 = 830 \, \text{m} \]
Now, dividing both sides by 2 to find \( d_2 \):
\[ d_2 = \frac{830}{2} = 415 \, \text{m} \]

### Step 4: Calculate the Total Distance Between the Hills
The total distance between the two hills is the sum of \( d_1 \) and \( d_2 \):
\[ \text{Total Distance} = d_1 + d_2 = 249 + 415 = 664 \, \text{m} \]

### Step 5: Calculate When the Third Echo Will Be Heard
The time for the first echo is \( \frac{3}{2} \) seconds and for the second echo is \( \frac{5}{2} \) seconds. The third echo will occur after another \( \frac{3}{2} \) seconds from the second echo.

Adding these times:
\[ t_3 = t_2 + \frac{3}{2} = \frac{5}{2} + \frac{3}{2} = \frac{8}{2} = 4 \, \text{s} \]

### Final Answers
- The distance between the hills is \( 664 \, \text{m} \).
- The time when the third echo will be heard is \( 4 \, \text{s} \).