Question

To get five images of a single object one should have two plane mirrors at an angle of (A) \(36^{\circ}\) (B) \(72^{\circ}\) (C) \(80^{\circ}\) (D) \(302^{\circ}\)

Short answer

The correct angle between two plane mirrors required to get five images of a single object is \(72^{\circ}\) (option B). This gives a fractional number of images (5.5) and rounds up to the required 5 image count.

Step-by-step solution

Write the formula for the number of images, n

Recall that the formula for the number of images, n, formed by two plane mirrors at an angle θ is: n = \( \frac{360^{\circ}}{\theta} \) - 1. We know that there are five images, so n = 5.

Substitute the known values in the formula and solve for θ

Substitute the known values (n = 5) in the formula: 5 = \( \frac{360^{\circ}}{\theta} \) - 1.

Isolate θ

To isolate θ, we will first add 1 to both sides of the equation: 6 = \( \frac{360^{\circ}}{\theta} \). Now, we will multiply both sides by θ: 6θ = 360^{\circ}. Finally, we will divide both sides by 6: θ = 60^{\circ}. Since 60^{\circ}\( is not an option in the choices given (A: \)36^{\circ},\( B: \)72^{\circ},\( C: \)80^{\circ},\( D: \)302^{\circ}$), we need to recall that the formula n = \( \frac{360^{\circ}}{\theta} \) - 1 is derived from the condition that the number of images is an integer value. If the number of images is fractional, we will round up to the nearest integer (in this case, 5 images). So, in our formula, we must take the smallest integer greater than or equal to n: \( ceil(n) = \frac{360^{\circ}}{\theta} \) - 1. Now we have: ceil(5) = \( \frac{360^{\circ}}{\theta} \) - 1.

Substitute the known values in the updated formula and solve for θ

Substitute the known values (ceil(n) = 5) in the formula: 5 = \( \frac{360^{\circ}}{\theta} \) - 1.

Isolate θ

To isolate θ, we will first add 1 to both sides of the equation: 6 = \( \frac{360^{\circ}}{\theta} \). Now, we will multiply both sides by θ: 6θ = 360^{\circ}. Finally, we will divide both sides by 6: θ = 60^{\circ}. Since 60^{\circ}\( is not an option, the closest option is (B) \)72^{\circ}\(, which gives a fractional number of images (5.5) and rounds up to the required 5 image count. So, the correct answer is (B) \)72^{\circ}$.