A tower 7 metres high stands on top of building which is 9 metres high. An observer at the bottom of the building notices that, as she walks away from the building, the angle theta which the tower subtends at her eyes seems to increase in size for a certain distance and then to decrease; determine what position of x maximises the angle theta.

Question

A tower 7 metres high stands on top of building which is 9 metres high. An observer at the bottom of the building notices that, as she walks away from the building, the angle theta which the tower subtends at her eyes seems to increase in size for a certain distance and then to decrease; determine what position of x maximises the angle theta.
Solve problem by using both a trigonometric and a geometric approach.

Steve · Answer

set up a diagram. You can see that if the angle to the top of the building is a, then

x/9 = cot(a)
x/16 = cot(a+θ)

θ = arctan(x/9) - arctan(x/16)
dθ/dx = 0 when x=12

as shown here:

https://www.wolframalpha.com/input/?i=arctan%28x%2F9%29+-+arctan%28x%2F16%29