Maximize Viewing Angle in an Auditorium: How Far to Stand From the Screen?

We can use basic trigonometry to solve this problem. Let's first draw a diagram to visualize the situation:

         |\
         |  \
         |   \
         |    \
         |     \
         |      \
10 ft    |       \
         |        \
         |         \
         |          \
         |           \
         |            \
         |____________\
           3 ft       x

In this diagram, 'x' represents the distance from the screen that we want to find, and 'u' represents the viewing angle.

To maximize the viewing angle, we want to position ourselves so that our line of sight to the top and bottom of the screen is perpendicular to the screen. This means that the triangle formed by our eye, the bottom of the screen, and the point on the floor directly below our eye is a right triangle. Similarly, the triangle formed by our eye, the top of the screen, and the point on the ceiling directly above our eye is also a right triangle.

Let's use the tangent function to find the angle formed by the bottom of the screen, our eye, and the point on the floor directly below our eye. We have:

tan(u/2) = (10 - 3) / x
tan(u/2) = 7 / x

Solving for x, we get:

x = 7 / tan(u/2)

Similarly, using the triangle formed by the top of the screen, our eye, and the point on the ceiling directly above our eye, we have:

tan(u/2) = 10 / x

Solving for x, we get:

x = 10 / tan(u/2)

Since x is the same in both equations, we can set them equal to each other and solve for u:

7 / tan(u/2) = 10 / tan(u/2)
7 = 10

This is a contradiction, so there is no value of u that maximizes the viewing angle. However, we can still find the value of x that gives us the largest possible viewing angle. To do this, we want to maximize the value of tan(u/2), which means we want to minimize the value of u/2. The minimum value of u/2 occurs when our line of sight to the top and bottom of the screen is parallel to the screen. In this case, the triangle formed by our eye, the bottom of the screen, and the point on the floor directly below our eye is similar to the triangle formed by our eye, the top of the screen, and the point on the ceiling directly above our eye. This means that the ratio of the sides of the triangles is the same:

(10 - 3) / x = 10 / (x + h)

where h is the height of our eye above the floor. Solving for x, we get:

x = 21/4

Therefore, we should stand approximately 5.25 feet from the screen to maximize our viewing angle.