Find the Length of a CarpetDate: 4/22/96 at 8:18:2 From: Stefan Isberg Subject: Find the Length of a Carpet You have a rectanguler room with sides of 3 and 5 meters. You put a carpet on the diagonal (not exactly, of course!) The carpet's corners must touch each wall in the room, so the problem will not be as easy as it seems. We have tried to illustrate how the carpet shall lie. The carpet lies approximately on the diagonal. 5 meter !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! \ ! ! \ ! ! \ ! 3 meter !\ ! ! \ ! ! \ ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! The carpet's width is 1 meter (The carpet is rectanguler) . What is the exact length of this carpet? Stefan and Anders, Math students at Umea University. Date: 5/3/96 at 4:15:3 From: Dr. Alain Subject: Re: Find the Length of a Carpet Draw the room like this: (-5,0) (0,0) ________________________________________ | | | | | | | | | | | | | | | | | | |________________________________________| (-5,-3) (0,-3) Say the carpet hits the top line at (-a,0) and the right line at (0,-b). Then it must hit the left wall at (-5, -3 + b), and hit the bottom wall at (-5 + a, -3). We know the width of the carpet is 1, so: a^2 + b^2 = 1. We also have that the carpet is rectangular, so the line from (-5 + a, -3) to (0,-b) makes a right angle with the line from (0,-b) to (-a,0). This is equivalent to the product of the slope of these two lines which is -1. The slope of the first line is m1= (3-b)/(5-a). The slope of the second line is m2= -b/a. So: m2*m1=-b/a*(3-b)/(5-a) = -1. Or b^2 - 3b = a^2 - 5a. So we have (1) a^2 + b^2 =1. (2) b^2 - 3b = a^2 - 5a. These two equations are enough to find a and b. These equations bring up a fourth degree polynomial equation. It is possible to find the exact solution to such an equation (when the solution exists) but the formula for fourth degree polynomials is quite horible. The best way to treat this polynomial equation is with numerical methods such as the Newton-Raphson algorithm. From the endpoints of the carpet we can find its length, L: L^2 = (-5 + a)^2 + (3 - b)^2. Now we have: L^2 = 25 - 10a + a^2 + 9 - 6b + b^2. Once you found a and b you can find L, which is the answer to your question. -Doctor Alain, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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