Question

A movie theater has a 24​-foot-high screen located 6 feet above your eye level. If you sit x feet back from the​ screen, your viewing​ angle, theta​, is as given below. theta equals tangent Superscript negative 1 Baseline StartFraction 30 Over x EndFraction minus tangent Superscript negative 1 Baseline StartFraction 6 Over x EndFraction Find the viewing​ angle, in​ radians, at distances of 5​ feet, 10​ feet, 15​ feet, 20​ feet, 25 feet.

Answer 1

The viewing angles are as follows:

For x=5 feet, θ = 0.529 radians

For x=10 feet, θ = 0.708 radians

For x=15 feet, θ = 0.726 radians

For x=20 feet, θ = 0.691 radians

For x=25 feet, θ = 0.640 radians

Explanation:

The viewing angle is given as:

θ = tan⁻¹(30/x) - tan⁻¹ (6/x)

where x is the distance between you and the screen.

The question is asking us to find the viewing angle θ at various distances. The distance value needs to be substituted in the above equation in place of x. So,

For x=5 feet:

θ = tan⁻¹(30/5) - tan⁻¹ (6/5)

= 1.4056 - 0.8761

θ = 0.529 radians

For x = 10 feet:

θ = tan⁻¹(30/10) - tan⁻¹ (6/10)

= 1.249 - 0.540

θ = 0.708 radians

For x = 15 feet:

θ = tan⁻¹(30/15) - tan⁻¹ (6/15)

= 1.107 - 0.380

θ = 0.726 radians

For x = 20 feet:

θ = tan⁻¹(30/20) - tan⁻¹ (6/20)

= 0.983 - 0.291

θ = 0.691 radians

For x = 25 feet:

θ = tan⁻¹(30/25) - tan⁻¹ (6/25)

= 0.876 - 0.235

θ = 0.640 radians