Multidimensional Calculus and Vector Geometry
Date: 02/09/99 at 05:42:46 From: Mathias Thomssen Subject: Multidimensional calculus, gradients, directional derivates Hello! I have trouble solving the following problem. Can you please help with a solution or give me a tip? At vertical trial borings from the points A, B, and C on the horizontal ground surface, iron ore was found at 150, 125 and 160 meters depth, respectively. The points form a right-angle triangle with the right angle in B. The distances AB and BC is 50 meters. The upper side of the deposit can be approximated with a plane. At a distance of maximum 50 meters from B, what point is the best to bore to find iron ore at the smallest depth possible? Also, find this depth. (The deposit is assumed to be spread under the whole area.) I appreciate your assistance! Best regards, Mathias Thomssen
Date: 02/09/99 at 13:35:12 From: Doctor Mitteldorf Subject: Re: Multidimensional calculus, gradients, directional derivates Dear Mathias, Multidimensional calculus is a tricky and difficult subject. Fortunately, you can solve this problem with more mundane analytic geometry. Set up a coordinate system on the plane of the earth's surface, with B as the origin and A at (50,0,0) and C at (0,50,0). We're told the top surface of the lode is a plane, and it passes through the three points (0,0,-125) and (50,0,-150) and (0,50,-160). Let's get an equation for this plane in the form z = ax + by + c. From the point B, we know c = -125. A tells us that a = -0.5, and C tells us that b = -.7. So the plane of the top surface is z = -125 - 0.5x - 0.7y. One way to take it from here is to use differential calculus to find a maximum value of z (closest to surface) on the circle x^2 + y^2 = 2500 (which represents the section of land that is no more than 50 meters from B.) So you'd be maximizing the function z = -0.5x + 0.7sqrt(2500-x^2) - 125. (How do I know to use a "+" sign in front of the square root?) But a simpler alternative is to use a little vector geometry to find the best direction. Moving toward +x from B makes things worse, so you want to move toward -x. Moving toward -y is also indicated. Since moving 1 meter toward -x gets you .5 meters of altitude, while moving 1 meter toward -y gets you .7 meters of altitude, your best bet is to move in the direction (-5,-7). (This sounds too simple to be true, but it really does work out that way.) So you need only find a point in the 3d quadrant of the circle x^2 + y^2 = 2500 where x and y are in the ratio 5:7. Start digging. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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