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/