The first resonance length of a resonance tube is `40 cm` and the second resonance length is `122 cm` . The third resonance length of the tube will be

Options

A. `(a)200 cm`
B. `(b)202 cm`
C. `(c)203 cm`
D. `(d)204 cm`

Explanation

To find the third resonance length of a resonance tube given the first and second resonance lengths, we can follow these steps:

### Step 1: Understand the relationship between resonance lengths
The resonance lengths in a tube are related to the wavelength of the sound wave. The first resonance length (L1) corresponds to a length of λ/2, the second resonance length (L2) corresponds to a length of λ, and so on.

### Step 2: Identify the given resonance lengths
We are given:
- First resonance length (L1) = 40 cm
- Second resonance length (L2) = 122 cm

### Step 3: Calculate the wavelength (λ)
The difference between the second and first resonance lengths gives us the wavelength:
\[ \text{Wavelength} \left( \lambda \right) = 2 \times (L2 - L1) \]
Substituting the values:
\[ \lambda = 2 \times (122 \, \text{cm} - 40 \, \text{cm}) = 2 \times 82 \, \text{cm} = 164 \, \text{cm} \]

### Step 4: Calculate the third resonance length (L3)
The third resonance length can be found by adding λ/2 to the second resonance length:
\[ L3 = L2 + \frac{\lambda}{2} \]
Substituting the values:
\[ L3 = 122 \, \text{cm} + \frac{164 \, \text{cm}}{2} = 122 \, \text{cm} + 82 \, \text{cm} = 204 \, \text{cm} \]

### Final Answer
The third resonance length of the tube is **204 cm**.

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