Have you ever wondered why some rows at the movie theater fill up super quickly or have a different price? Or why did Sheldon (in The Big Bang Theory TV series) start changing seats and doing soundproofing before the movie started? See the video below.
Since they don't have Sheldon's superpowers, my G10 students used math to find the best row given the cinema characteristics: screen size, the distance between rows, inclination angle, etc. Keep reading to find out how they did it.
Spoiler alert!: You can find the original project on the Stewartmath website as part of one of the textbook resources called Discovery Project 10: Where to Sit at the Movies.
Individually, people may prefer particular seats, but we may agree that the seats somewhere in the middle are the most popular for the majority. Or, as Stewart, Redlin, and Watson formally say it:
To best view a painting or a movie, the viewing angle must be as large as possible. If the painting or movie screen is above eye level, being too far away or too close results in a small viewing angle and a poor viewing experience.
Let's Jump to the project.
You can find the original project on the Stewartmath website.
Question 1 [Without inclination]
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, where the angle θ subtended by the screen at your eyes is a maximum. Suppose your eyes are 4 ft above the floor, as in Figure 2, and you sit at a distance x from the screen.
Question 1 Part a.
Relationship between the angle of vision and the distance from the screen. Show that
Solution Q1 Part a
Keypoint: Since the observer's vision line is not straight, we can draw a horizontal line at 4 ft.
Let's use this on the triangles containing β and α
Question 1 Part b. Let's organize that function of the angle theta a bit.
To do this, we can use the addition rule of the tangent as follows:
Question 1 Part c. Let's find out where to sit.
Use a graphing device to graph the angle (θ) as a function of the distance from the screen (x).
What value of x maximizes θ?
In which row should you sit?
What is the viewing angle in this row?
Solution Q1 Part c
This question graph was done using Desmos (remember to set your device to degrees since all the angles in this project are on degrees). As you can see on the graph below, when x=13, we have the maximum value for θ. This angle is 40.32 degrees. And it's achieved when you sit in the third row, 7ft from the screen to the first row, plus 6ft from the first two rows (see figure 2).
If you are a little more advance and have already seen derivates as a tool to find the minimum and maximum of a function, you can see on the graph that at 12.961, the derivate is zero. This represents the location of the max of the function in this scenario.
But actual movie theaters have some inclination
Question 2 [With inclination]
Now suppose that starting with the first row of seats, the floor of the seating area is inclined at an angle of α=25 degrees above the horizontal, and the distance you sit up the incline is x, as shown in the figure.
Download the answers for this project along with the subsequent questions below:
References
Stewart, J. (2014). Precalculus Mathematics for Calculus (7th ed.). Cengage Learning.
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