In a stationary wave that forms as a result of reflection of waves from an obstacle, the ratio of the amplitude at an antinode to the amplitude at node is 6. What percentage of energy is transmitted?

Explanation

To solve the problem step by step, we will analyze the relationship between the amplitudes at the antinode and node, and then determine the percentage of energy transmitted.

### Step 1: Understand the relationship between amplitudes
Given that the ratio of the amplitude at an antinode (A_antinode) to the amplitude at a node (A_node) is 6, we can express this as:
\[ \frac{A_{\text{antinode}}}{A_{\text{node}}} = 6 \]

### Step 2: Define incident and reflected amplitudes
Let:
- \( A_I \) = amplitude of the incident wave
- \( A_R \) = amplitude of the reflected wave

At the antinode, the total amplitude is the sum of the incident and reflected amplitudes, and at the node, the total amplitude is the difference. Therefore, we can write:
\[ \frac{A_I + A_R}{A_I - A_R} = 6 \]

### Step 3: Set up the equation
From the ratio, we can cross-multiply:
\[ A_I + A_R = 6(A_I - A_R) \]
Expanding this gives:
\[ A_I + A_R = 6A_I - 6A_R \]
Rearranging terms results in:
\[ A_R + 6A_R = 6A_I - A_I \]
\[ 7A_R = 5A_I \]

### Step 4: Solve for the ratio of reflected to incident amplitude
From the equation \( 7A_R = 5A_I \), we can express the ratio of the reflected amplitude to the incident amplitude:
\[ \frac{A_R}{A_I} = \frac{5}{7} \]

### Step 5: Calculate the energy ratio
The energy associated with a wave is proportional to the square of its amplitude. Therefore, the ratio of the reflected energy \( E_R \) to the incident energy \( E_I \) can be expressed as:
\[ \frac{E_R}{E_I} = \left(\frac{A_R}{A_I}\right)^2 = \left(\frac{5}{7}\right)^2 = \frac{25}{49} \]

### Step 6: Calculate the percentage of reflected energy
To find the percentage of reflected energy, we multiply by 100:
\[ \text{Percentage of reflected energy} = \frac{25}{49} \times 100 \approx 51.02\% \]

### Step 7: Determine the percentage of transmitted energy
The transmitted energy is the remaining energy after reflection:
\[ \text{Percentage of transmitted energy} = 100\% - \text{Percentage of reflected energy} \]
\[ \text{Percentage of transmitted energy} = 100\% - 51.02\% \approx 48.98\% \]

### Final Answer
Thus, the percentage of energy transmitted is approximately:
\[ \boxed{49\%} \]