In this video I go over another Applied Project problem and go about answering the problem of determining the best seat to sit at when at the movie theatre. The applied project problems are from the end of each chapter in my Calculus textbook and they represent problems in everyday life that can be solved using Calculus. In this particular problem, the best seat to sit at is to maximize the viewing angle when watching a movie to ensure that the largest part of the movie screen is being seen at the most direct path. This problem is very useful for whenever you are contemplating which seat you should take, just like Rebecca Black ;)
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Applied Project: Where to Sit at the Movies
A movie theater has a screen that is positioned 10 ft off the floor and is 25 ft high. The first row of seats is placed 9 ft from the screen and the rows are set 3 ft apart. The floor of the seating area is inclined at an angle of α = 20° above the horizontal and the distance up the incline that you sit is x. The theater has 21 rows of seats, so 0 ≤ x ≤ 60. Suppose you decide that the best place to sit is in the row where the angle ϴ subtended by the screen at your eyes is a maximum. Let’s also suppose that your eyes are 4 ft above the floor, as shown in the figure.
(1) Show that:
where:
and:
(2) Use a graph of ϴ as a function of x to estimate the value of x that maximizes ϴ. In which row should you sit? What is the viewing angle ϴ in this row?
(3) Use your computer algebra system to differentiate ϴ and find a numerical value for the root of the equation dϴ/dx = 0. Does this value confirm your result in Problem 2?
Texas Instruments Free Trial on their TI-NSpire CAS Software
This confirms our results from problem 2.
(4) Use the graph of ϴ to estimate the average value of ϴ on the interval . Then use your CAS to compute the average value. Compare with the maximum and minimum values of ϴ.
