bonanova Posted August 20, 2009 Report Share Posted August 20, 2009 A couple turns of the century ago my grandfather homesteaded a few hundred acres of wheat land in NW North Dakota. Today that land is famous not for its wheat but for its oil, buried in shale, miles under the ND prairies. This gives rise to a geometry challenge, and this time it's 3-dimensional. An oil drill [unfortunately not on Grandpa's land] found oil-bearing shale at an underground location that was determined to be exactly 28,000 feet from one corner of a [rectangular] wheat field, 24,000 feet from the opposite corner and 36,000 feet from a third corner. How far is the underground location from the fourth corner of the field? Note: we're not looking for the location on the surface where the drilling began. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 So all we are looking for is a tetrahedron with given sides from the oil shale vertex, and with another vertex which is the corner of a square. I am just going out for a lunch date. Of course I am interested in this interesting 3D question, but my date might not be amused! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 16,000. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 I assumed the real dept of the shale as zero (from drilling point). This is not realistic, but I thought "why not?". Then I found the result as 8000 feet??? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 8000 feet though that seems small Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 I agree with burn and nobody- Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 (edited) It seems small, but it's 8000 Attached is a small pdf file that shows the solution. Summarizing, it boils down to setting up a system of sphere equations which are centered at the four corners of the rectangle and solving for the intersection point Edited August 20, 2009 by tpaxatb Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 use the op we make a rectangle weth sides w & h the spot is x in front of, y to the right of a z below one corner p-thiggy tells us (1) sqrt(x2+y2+z2)=24,000 (2) sqrt((w-x)2+(h-y)2+z2)=28,000 (3) sqrt((w-x)2+y2+z2)=36,000 (4) sqrt(x2+(h-y)2+z2)=A square both sides of each equation and subtract (1) from (3) and (4) from (2) from there the answer is easy to find. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 32000 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 20, 2009 Report Share Posted August 20, 2009 Huh. I just checked my work and noticed: That while I reversed the signs of the right side of equations 5,6, and 7...the corrected equations still gave the same answer... (corrected) spheres.pdf Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted August 21, 2009 Author Report Share Posted August 21, 2009 Nice work all. Note that nobody's incorrect assumption leads nevertheless to the correct answer. And the pretty relationship among the four distances, in both 2-D and 3-D. We're waiting for word of a drill on G-pa's wheat field. Quote Link to comment Share on other sites More sharing options...
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bonanova
A couple turns of the century ago my grandfather homesteaded a few hundred acres of wheat land in NW North Dakota.
Today that land is famous not for its wheat but for its oil, buried in shale, miles under the ND prairies.
This gives rise to a geometry challenge, and this time it's 3-dimensional.
An oil drill [unfortunately not on Grandpa's land] found oil-bearing shale at an underground location that
was determined to be exactly 28,000 feet from one corner of a [rectangular] wheat field, 24,000 feet
from the opposite corner and 36,000 feet from a third corner.
How far is the underground location from the fourth corner of the field?
Note: we're not looking for the location on the surface where the drilling began.
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