A 4 by 4 picture hangs on a wall such that its bottom edge is 2 ft above your eye level. How far back from the picture should you stand, directly in fron of the picture under the maximum angle?
Click HERE to investigate this problem using GSP.
Consider the following diagram.
In order to maximize angle ø consider the following equations in terms of x
.
Solve for x, substitue
into, simplify, and
solve for ß.
The resulting equation is ß = arctan(3 tan µ)
Since ø = ß - µ, substituting ß = arctan(3 tan µ) we have ø = arctan(3 tan µ) - µ.
To maximize ø, we simply take the derivative of ø = arctan(3 tan µ) - µ and set it equal to zero.
We can now solve for µ.
Keeping the relevant domain of µ, 
, in
mind, we find that when ø is a maximum
. Hence to find the maximum
viewing angle ø = arctan(3 tan µ) - µ, simply
substitute and evaluate.
So the maximum viewing angle is 