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Best movie theater seats

A movie theater has a screen that is positioned 10 feet off the floor and is 25 feet high. The first row of seats is placed 9 feet from the screen and the rows are 3 feet apart. The floor of the seating area is inclined at an angle above the horizontal and the distance up the incline that you sit is x. The theater has 21 rows of seats.

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Suppose you decide that the best place to sit is in the row where the angle subtended by the screen at your eyes is a maximum. Let's also suppose that your eyes are 4 feet above the floor.

Show that: ---insert the second file image here---

Use a graph of theta as a function of x to estimate the value of x that maximizes theta.

In which row should you sit? What is the viewing angle theta in this row?


Solution Preview

The first half of the problem, showing a relationship between x and theta, involves using the cosine law and Pythagoras (which is just a special case of the cosine law) on a few carefully chosen triangles.
solution requires drawing out a picture - so as I try to explain, please take a piece of paper out and try drawing the pictures as we go to see where the various triangles are.

Ok - first of all I started by drawing out the picture for myself. I marked out where the various lengths were - the 25, 10, 9, and 4. The picture should be like the one you included - but bigger, and a little more abstract. (There really isn't any benefit in sketching little seats for example. My person is represented by a vertical line segment of length 4).
In solving this problem, I started by looking at the answer: the arccos suggested that the law of cosines might be involved:

a^2 + b^2 = c^2 + 2ab cos(theta)
where theta is the angle opposite to side c and the a,b,c are the lengths of the sides ...

Solution Summary

This explains how to find the best place to sit in a theater based on the angles and distances involved.