Chapter 6.4, Problem 54E

Viewing Angle of an Observer While visiting a museum, an observer views a painting that is 3 ft high and hangs 6 ft above the ground. Sec the figure. Assume her eyes are 5 ft above the ground, and let x be the distance from the spot w here she is standing to the wall displaying the painting.

(a) Show that θ. the viewing angle subtended by the painting, is given by

$\theta ={\tan }^{-1}\left(\frac{4}{x}\right)-{\tan }^{-1}\left(\frac{1}{x}\right).$

(b) Find the value of x to the nearest hundredth for each value of θ.

(i) $\theta =\frac{\pi }{6}$

(ii) $\theta =\frac{\pi }{8}$

(c) Find the value of θ to the nearest hundredth for each value of x.

(i) x = 4

(ii) x = 3