1.) An oil drilling rig located 14 miles off of a straight coastline is to be co
ID: 3190946 • Letter: 1
Question
1.) An oil drilling rig located 14 miles off of a straight coastline is to be connected by a pipeline to a refinery 10 miles down the coast from the point directly opposite the drilling rig. Laying pipe under water costs 500,000 dollars per mile. Laying pipe on land costs 300,000 dollars per mile. What combination of underwater and land-based pipe will minimize the total cost of the pipeline? 2.) a.)The US Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) is does not exceed 108 inches, What are the dimensions of the cylindrical box (circular cross section, closed ends) with largest volume that can be shipped in the US mail? b.) explain how you know for sure that the dimensions you found give the box with the largest rather than the smallest volume. c.)Students often describe their method for solving optimization problems as "set the derivative equal to zero and solve for x". Is the solution to an optimization problem always a point where the derivative is zero? If so, explain why. If not, explain why not.Explanation / Answer
A rig
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B ...C ...D refinery
we shall solve in general form & then substitute numerical values
let AB = w (14 mi)
BC = x
BD = y (10 mi)
cost ratio/mi water:land = k (5/3)
angle BAC = z
dist. over water = w•sec z,
dist. over land = y - w•tan z
cost C 8 k•sec z +(y - tan z)
dC/dz = k•secz•tan z - sec^2 z
= (ksin z - 1) / cos^2 z
for minima, sin z = 1/k
which yields x = w /v(k^2 - 1) = 14/sqrt((5/3)^2-1) = 10.5 mi
THIS MEANS THAT THE PIPIELINE MUST BE LAID TOTALLY UNDERWATER STRAIGHT TO THE REFINERY !
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