Chapter 6.1, Problem 107E

Observation of a Painting A painting 1 m high and 3 m from the floor will cut off an angle θ to an observer, where $\theta ={\tan }^{-1}\left(\frac{x}{x^2+2}\right),$

assuming that the observer is x meters from the wall where the painting is displayed and that the eyes of the observer are 2 m above the ground. (See the figure.) Find the value of θ for the following values of x. Round to the nearest degree.

(a) 1 (b) 2 (c) 3

(d) Derive the formula given above. (Hint: Use the identity for tan(θ + α). Use right triangles.)

(e) Graph the function for θ with a graphing calculator, and determine the distance that maximizes the angle.

(f) The concept in part (c) was first investigated in 1471 by the astronomer Regiomontanus. (Source: Maor, E., Trigonometric Delights. Princeton University Press.) If the bottom of the picture is a meters above eye level and the lop of the picture is b meters above eye level, then the optimum value of x is √ab meters. Use this result to find the exact answer to part (e).