How Should the Pipe be Laid?
Date: 03/25/99 at 01:10:53 From: Kristin Babin Subject: Minimizing cost from an oil rig to the shore Dear Dr. Math, Here are two calculus problems that I could not get. It seems as if one problem is giving extra information, and the other problem is just weirdly worded. Please help me with this if you can. 1. Oil from an offshore rig located 3 miles from the shore is to be pumped to a location on the edge of the shore that is 8 miles east of the rig (8 miles along the shore). The cost of constructing a pipe in the ocean from the rig to the shore is 1.5 times as expensive as the cost of construction on land. Determine how the pipe should be laid to minimize cost. When I tried to solve this problem, I kept coming up with the Pythagorean theorem. I know the problem is not that easy, but my answer was the square root of 73 miles. Is that right? 2. A fence that is 8 feet high stands 27 feet from the wall of a building. What is the length of the shortest straight beam that will reach to the side of the building from the ground outside the fence? Again, I used the Pythagorean theorem and got the square root of 793 feet. I really don't think that is the answer.
Date: 03/25/99 at 08:03:29 From: Doctor Jerry Subject: Re: Minimizing cost from an oil rig to the shore Hi Kristin, 1. If x is the distance from (the point of the shore closest to the oil rig) to (the point on the shore where the pipe comes ashore), then y^2 = 3^2+x^2, where y is the length of pipe in the ocean. The length of the pipe on the land is 8-x, right? So, the cost is C(x) = 1.5*sqrt(9+x^2) + 1*(8-x), where 0 <= x <= 8. If you set C'(x) equal to zero, you'll find x = 6/sqrt(5). Check C at this point, as well as at x = 0 and x = 8. 2. Let x be the distance from bottom of the fence to the point on the beam that rests on the ground. Let y be the distance on the building, from the ground to where the beam hits. From similar triangles, y/(27+x) = 8/x. So, you can write y in terms of x. The length of the beam is sqrt(y^2+(27+x)^2). Differentiate this w.r.t x and set equal to zero. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/
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