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Question C 03
At the outdoor summer movie at Stanley Park, the bottom of the screen is 2 meters above your eye level, and the screen is 6 meters tall. At what distance from the base of the screen is the visual angle occupied by the screen as large as possible?
HINT: There are several possible approaches to this problem but one approach is to define as the angle to the top of the screen and as the angle to the bottom of the screen and maximize their difference.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Maximize the difference of the two angles and with implicit differentiation
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
As indicated in the hint, we define as the angle to the top of the screen* and as the angle to the bottom of the screen. Then Maximizing the visual angle occupied by the screen is equivalent to maximizing the angle . As and are both functions of , we just need to calculate the derivative and find the critical . First with implicit differentiation we calculate
Note: It is a common mistake to interpret the angle as the angle that is below the angle and not the angle that is within the angle .
With the diagram on the right
we obtain So Similarly, we calculate Hence,
By setting we solve , as represents the distance and must be positive. To verify it is a maximum, we use the first derivative test
So the angle is maximized at .