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### Multidimensional Calculus and Vector Geometry

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Date: 02/09/99 at 05:42:46
From: Mathias Thomssen
Subject: Multidimensional calculus, gradients, directional derivates

Hello!

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.)

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/
```
Associated Topics:
College Calculus
College Higher-Dimensional Geometry
High School Calculus
High School Higher-Dimensional Geometry

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