Small 1. The screen in a theatre is 22 ft high and is positioned 10 ft above the

Question

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?

Explanation / Answer

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)]

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