MATH102 December 2014
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• QA 1 • QA 2 • QA 3 • QA 4 • QA 5 • QA 6 • QA 7 • QA 8 • QB 1 • QB 2 • QB 3 • QB 4 • QB 5 • QB 6 • QB 7 • QC 1 • QC 2(a) • QC 2(b) • QC 2(c) • QC 2(d) • QC 3 •
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.
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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
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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 .
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