# Determining the best place to sit at the movies based on a set of criteria.

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?

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#### Solution Preview

... a triangle. Notice that if theta = 90 then this just becomes Pythagoras (cos90=0).
Since 25^2 = 625 we can start to see the solution involves just drawing the right triangle with a and b labeled appropriately.
Secondly, a^2 and b^2 seem to be found from using Pythagoras - so we have to look for right angle triangles!
Using the answer as a guide, I figure that there must be two important right angle triangles as shown in my attached drawing.
Notice that the green triangle allows us to use the angle alpha to get the distance x into the equation. Hopefully it is clear where the side lengths for the blue and red triangles come from - think about what pieces make up each line segment. Then, using Pythagoras you can get what a^2 and b^2 are (same as in the answer) and finally using the law of cosines above with c=25 gives the answer. (You need to rearrange and solve for theta. Done. (The real trick is drawing the right picture - it may be useful to draw the picture also if you are ...