Question

help The screen in a theatre is 22 ft high and is positioned 10 ft above the floor, which is flat. The first row of seats is 7 ft from the screen and the rows are 3 ft apart. You decide to sit in the row where you get the maximum view, that is, where the angle theta subtended by the screen at your eyes is a maximum. Suppose your eyes are 4 ft above the floor, and you sit at a distance x from the screen. a) Show that Theta = arctan(28/x) - arctan(6/x) b) Use the subtraction formula for tangent to show that Theta = arctan(22x/(x^2) + 168)

Answer 1

 (a.) 
Let's say α is the angle that subtends from the top of the screen to horizontal eye-level. 
Let β be the angle that subtends from the bottom of the screen to horizontal eye-level. 

tanα = (22 + 10 - 4) / x = 28/x 
α = arctan(28/x) 

tanβ = (10 - 4) / x = 6/x 
β = arctan(6/x) 

Ɵ = α - β 
Ɵ = arctan(28/x) - arctan(6/x) 

(b.) 
tanƟ = tan(α - β) = (tanα - tanβ) / (1 + tanα tanβ) 
tanƟ = (28/x - 6/x) / [1 + (28/x)(6/x)] 
tanƟ = (22/x) / [1 + (168/x²)] 
tanƟ = 22x / (x² + 168) 
Ɵ = arctan[22x / (x² + 168)]