Small 1 The screen in a theatre is 22 ft high and is positio

Small 1. 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 subtended by the screen at your eyes is a maximum. Suppose your eyes are 4 ft above the floor, as in the figure, and you sit at a distance x from the screen 22 ft 10 ft 4 ft 3 ft 7 ft (a) Show that = tan i tan i (b) Use the subtraction formula for tangent to show that 22x tan (x2 + 168 (c) Use a graphing device to graph 0 as a function of x. What value of x maximizes 0? In which row should you sit? What is the viewing angle in this row?

Solution

1)

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

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

tan = (10 - 4) / x = 6/x
= arctan(6/x)

= -
= arctan(28/x) - arctan(6/x)

(b.)
tan = tan( - ) = (tan - tan) / (1 + tan tan)
tan = (28/x - 6/x) / [1 + (28/x)(6/x)]
tan = (22/x) / [1 + (168/x²)]
tan = 22x / (x² + 168)
= arctan[22x / (x² + 168)]