Rope between Two Poles

Date: 05/04/99 at 13:26:04
From: Fred Klock
Subject: Calculus

Dr. Math,

This is a problem in a calculus book. One of my 11th grade students 
has completed all of the math we offer at our high school. The 
following is a problem he has been having difficulty with and I can't 
quite figure it out for him.

Two poles, not necessarily the same height, are perpendicular to the 
ground: PQ and ST with perpendicular points Q and T. On the ground 
between Q and T is point R. There is a rope from P to R to S. Show 
that the shortest length of rope occurs when angle PRQ = angle SRT.

We tried similar triangles to create proportions and we tried using 
trig functions, but kept coming up with too many variables. We also 
tried the Pythagorean theorem and the fact that the total rope could 
be expressed as PR + RS = PS, etc.

Please do this for us and send the explanations.
Thank you very much.

Respectfully,
Fred Klock
Mahanoy Area High School

Date: 05/06/99 at 16:28:56
From: Doctor Fwg
Subject: Re: Calculus 140

Dear Mr. Klock,

This is an interesting problem. The algebra is a little involved (but 
there are a few "tricks" you can use to reach the solution without 
great difficulty - I think you will see what I mean if you go 
through the problem more than one time), so I will only outline it.  
If you have any trouble in verification or in understanding any of my 
explanation, write back and I will supply more detail.

SPECIFICATIONS

Constants:
a = above-ground perpendicular height of first pole
b = above-ground perpendicular height of second pole
L = horizontal distance, at ground level, between poles   

Variables:
x = distance between point "R" and second pole, at ground level
(L - x) = distance between point "R" and first pole, at ground level
R1 = length of rope from top of first pole to point "R" on ground 
R2 = length of rope from top of second pole to point "R" on ground
B1 = angle subtended between ground and rope between top of first 
     pole and point "R" on ground 
B2 = angle subtended between ground and rope between top of second 
     pole and point "R" on ground
D = R1 + R2

Now, what you want to "prove" is that when (R1 + R2) is a minimum, 
angle B1 = B2. First, you might want to write R1 and R2 as a function 
of x, using the Pythagorean theorem, then take the first derivative of 
the sum (R1 + R2), set that equal to zero, and then solve for x and 
(L-x) as functions of a, b, and L.

Note:
a, b, and L are all constants, 
R1^2 = a^2 + (L - x)^2, and
R2^2 = b^2 + x^2.

If you set:
D = R1 + R2 = [a^2 + (L - x)^2]^(1/2) + [b^2 + x^2]^(1/2),
then find: dD/dx, then set dD/dx = 0, things might go easier.

After you find x and (L - x) as functions of a, b, and L, you will be 
able to show that B1 = B2, by showing that Tan[B1] = Tan[B2]. All the 
other trig functions will also be equal for B1 and B2 but the tan 
function will probably be easier to work with in this problem. That's 
it. Good luck, and do not hesitate to write back if any clarification 
is needed.

- Doctor Fwg, The Math Forum
http://mathforum.org/dr.math/