To get the best view of the statue of liberty, you should be at the position where angle theta is a maximum?

Question

If the statue stands 92 meters high, including the pedestal, which is 46 meters high, how far from the base should you be? Hint: Find a formula for angle theta in terms of your distance from the base. Use this function to maximize theta, noting that theta is greater than or equal to 0 and less than or equal to pi/2.

Answer 1

You are missing a constraint in your problem. If your only constraints are
- theta must be less than or equal to pi/2
- you want to maximize theta
then theta = pi/2.

So how far should you be? 0 meters.

Answer 2

Construct a right triangle ABC as such:

The right angle A is the foot of the pedestal;
The vertical side AB represents the statue (with pedestal), length 92; B is the point representing the head of the statue.
The base of the triangle represents the distance between you (point C) and the foot of the pedestal.

Make point D the midpoint of segment AB (the top of the pedestal). Now draw the segment CD (from you to the top of the pedestal). You want to maximize angle
You are looking for x = the length of the base, i.e. the length of segment AC. The angle you are trying to maximize is...

= arccotg(x/92) - arccotg(x/46).

Derive, using d(arccotg(x))/dx = -1/(1+x^2):


After a bit of algebra, you find that this is equal to zero when x = sqrt(92*46) = 46sqrt(2) = about 65 metres. The resulting angle is arccotg(sqrt(1/2)) - arccotg(sqrt(2)) = 54.74° - 35.26° = 19.47°.