Rope between Two PolesDate: 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/ |
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