Optimization Problem

Question

Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. If R is chosen to minimize the length of the rope, Find the ratio

𝜃1𝜃2

in terms of a and b where a is the length of PQ and b is the length of ST.

Answer 1

If these are the angles of elevation from R to P and S then the angles are the same.

Reason :

suppose the distance between the posts on the ground is L, and point R is x from Q.

Then if s = length of rope.

s = (a^2 + x^2) ^ (1/2) + (b^2 + (L-x)^2)^(1/2)

We choose x so that ds/dx = 0

ds/dx = (1/2) (a^2 + x^2) ^(-1/2) (2x) + (1/2) (b^2 + (L-x)^2) ^(-1/2) (2 (L-x) (-1))

ds/dx = 0 gives a/x = b/L-x after some nasty algebra

These are the tangents of the two angles , which are clearly equal, meaning the angles must be congruent.

x = aL/(a+b)