Towers Task - Fostering a Geometric Argument
Two vertical towers (PQ and ST) are secured by a steel cable. The cable goes from the top of one tower (P), to a point on the ground between the towers (R), to the top of the other tower (T). Assume that the height of tower is double the height of the other. Find the position of R that requires the least amount of cable. Calculate the minimal cable length. (Adapted from Calculus 4th Ed. (Stewart, p. 285, item 48)).
Reflect tower T over segment QS (i.e., the ground). What (if anything) do you notice about the position of R that minimizes total cable length? Justify any patterns that you find.