Two mirrors at an angle `theta^(@)` produce 5 images of a point. The number of images produced when `theta` is decreased to `theta^(@) - 30^(@)` is

Options

A. 9
B. 10
C. 11
D. 12

Explanation

To solve the problem step by step, we will use the formula for the number of images produced by two mirrors set at an angle \( \theta \):

### Step 1: Understand the formula for the number of images
The formula for the number of images \( N \) produced by two mirrors at an angle \( \theta \) is given by:
\[ N = \frac{360^\circ}{\theta} - 1 \]

### Step 2: Set up the equation for the initial condition
We know that when the angle is \( \theta \), the number of images produced is 5. Therefore, we can set up the equation:
\[ 5 = \frac{360^\circ}{\theta} - 1 \]

### Step 3: Solve for \( \theta \)
Rearranging the equation gives:
\[ 5 + 1 = \frac{360^\circ}{\theta} \]
\[ 6 = \frac{360^\circ}{\theta} \]
Now, multiplying both sides by \( \theta \) and then dividing by 6 gives:
\[ \theta = \frac{360^\circ}{6} = 60^\circ \]

### Step 4: Determine the new angle
According to the problem, the angle is decreased by \( 30^\circ \). Therefore, the new angle \( \theta' \) is:
\[ \theta' = 60^\circ - 30^\circ = 30^\circ \]

### Step 5: Calculate the number of images for the new angle
Now we will use the formula again to find the number of images produced at the new angle \( \theta' \):
\[ N' = \frac{360^\circ}{30^\circ} - 1 \]
Calculating this gives:
\[ N' = 12 - 1 = 11 \]

### Conclusion
The number of images produced when the angle is decreased to \( 30^\circ \) is \( 11 \).

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