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

The optimal viewing angle theta in a theater can be calculated using the difference between the arctangents of the screen's top and bottom from your eye level, simplified into a single arctangent using the subtraction formula for tangent.

Step-by-step explanation:

To solve for theta, you need to consider that theta is the angle at which you look at the screen, which can be broken into two separate angles corresponding to the top and bottom of the screen. If your eyes are 4 ft above the floor, the top of the screen would be 22 ft + 10 ft above the floor but since your eye level is taken into account, it becomes 22 + 10 - 4 = 28 ft. Similarly, the bottom of the screen is at 10 ft above the floor and therefore from your eye level, it is at 10 - 4 = 6 ft.

The angle for any point on the screen can be found using the arctangent, which gives the angle whose tangent is the ratio of the opposite side to the adjacent side in a right-angled triangle. Using this, the angle subtended by the top of the screen Thetay can be expressed as arctan(28/x) and the bottom Thetar as arctan(6/x). Therefore, the angle theta you are looking to maximize is the difference between these two angles, which is Theta = arctan(28/x) - arctan(6/x).

Using the subtraction formula for the tangent, tan(Theta1 - Theta2) = (tan Theta1 - tan Theta2) / (1 + tan Theta1tan Theta2), we can then transform the difference of arctangents into a single arctangent Theta = arctan((tan(28/x) - tan(6/x)) / (1 + tan(28/x)tan(6/x))). When you do the math and simplify, you end up with Theta = arctan(22x/(x^2) + 168), as long as x2 > 0.

Answer 2

(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)]