A 1.80-m-tall person stands 4.20 m from a convex mirror and notices that he looks precisely half as tall as he does in a plane mirror placed at the same distance. What is the radius of curvature of the convex mirror? (Assume that θ ≈ θ .) [Hint: The viewing angle is half.]

Two plane mirrors are facing each other 2.2 m apart as in Fig. 32–60. You stand 1.5 m away from one of these mirrors and look into it. You will see multiple images of yourself. (a) How far away from you are the first three images of yourself in the mirror in front of you? (b) Are these first three images facing toward you or away from you?

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A triangular prism made of crown glass (n = 1.52) with base angles of 26.0° is surrounded by air. If parallel rays are incident normally on its base as shown in Fig. 32–66, what is the angle Φ between the two emerging rays?

We wish to determine the depth of a swimming pool filled with water by measuring the width (x = 5.20m) and then noting that the bottom edge of the pool is just visible at an angle of 13.0° above the horizontal as shown in Fig. 32–61. Calculate the depth of the pool.

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The critical angle of a certain piece of plastic in air is θ_C = 35.8°. What is the critical angle of the same plastic if it is immersed in water?

The label on a laser says it produces light of wavelength 670 nm. The laser beam passes through a block of plastic for which n = 1.57. What is the wavelength of the light inside the plastic?

When light passes through a prism, the angle that the refracted ray makes relative to the incident ray is called the deviation angle δ, Fig. 32–64. Show that this angle is a minimum when the ray passes through the prism symmetrically, perpendicular to the bisector of the apex angle Φ, and show that the minimum deviation angle, δ_m, is related to the prism’s index of refraction n by

$n=\frac{\sin \frac{1}{2}(\varphi +{\delta }_m)}{\sin \varphi /2}.$

[Hint: For θ in radians, $\left(d/d\theta \right)\left({\sin }^{-1}\theta \right)=1/\sqrt{1-{\theta }^2}$.]