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An oil refinery is located on the north shore of a river at point A in the diagram below. They wish to lay a pipe to move their refined product to the railway yard, located on the south shore at point D, to get the oil to market. The river is 550 m wide (so the distance from point A to point B is 550 m), and the railway yard is 1340 m downriver (so the distance from point B to point D is 1340 m). It costs $1400 per m to lay pipe underwater, and $1000 per m to lay pipe over land. At what point C on the south shore of the river should they aim the underwater pipe in order to minimize the cost of the piping? Give your answer as the distance from point B. A river B C To solve this problem, you will apply the Closed Interval Method. The objective function, the function you want to minimize, is the cost. If we let x be the distance from point B to point C, then C(x) = D

An oil refinery is located on the north shore of a river at point A in the diagram below. They wish to lay a pipe to move their refined product to the railway yard, located on the south shore at point D, to get the oil to market. The river is 550 m wide (so the distance from point A to point B is 550 m), and the railway yard is 1340 m downriver (so the distance from point B to point D is 1340 m). It costs $1400 per m to lay pipe underwater, and $1000 per m to lay pipe over land. At what point C on the south shore of the river should they aim the underwater pipe in order to minimize the cost of the piping? Give your answer as the distance from point B. A river B C To solve this problem, you will apply the Closed Interval Method. The objective function, the function you want to minimize, is the cost. If we let x be the distance from point B to point C, then C(x) = D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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An oil refinery is located on the north shore of a river at point A in the diagram below. They wish to lay a pipe to move their refined product to the railway yard, located on the south shore at point D, to get the oil to market. The river is 550 m wide (so the distance from point A to point B is 550 m), and the railway yard is 1340 m downriver (so the distance from point B to point D is 1340 m). It costs $1400 per m to lay pipe underwater, and $1000 per m to lay pipe over land. At what point C on the south shore of the river should they aim the underwater pipe in order to minimize the cost of the piping? Give your answer as the distance from point B. A river B с To solve this problem, you will apply the Closed Interval Method. The objective function, the function you want to minimize, is the cost. If we let I be the distance from point B to point C, then C(x) = The closed interval over which you want to minimize this function is: x E D The Closed Interval Method, an application of the Extreme Value Theorem, applies in this case because C(x) is on this closed interval. Find the critical point of C(x) in the interior of this interval. x = Evaluate C(x) at the left endpoint: C = the critical point: C = the right endpoint: C = Based on the above, we may conclude that the distance from B to C that minimizes the cost is m
Transcribed Image Text:An oil refinery is located on the north shore of a river at point A in the diagram below. They wish to lay a pipe to move their refined product to the railway yard, located on the south shore at point D, to get the oil to market. The river is 550 m wide (so the distance from point A to point B is 550 m), and the railway yard is 1340 m downriver (so the distance from point B to point D is 1340 m). It costs $1400 per m to lay pipe underwater, and $1000 per m to lay pipe over land. At what point C on the south shore of the river should they aim the underwater pipe in order to minimize the cost of the piping? Give your answer as the distance from point B. A river B с To solve this problem, you will apply the Closed Interval Method. The objective function, the function you want to minimize, is the cost. If we let I be the distance from point B to point C, then C(x) = The closed interval over which you want to minimize this function is: x E D The Closed Interval Method, an application of the Extreme Value Theorem, applies in this case because C(x) is on this closed interval. Find the critical point of C(x) in the interior of this interval. x = Evaluate C(x) at the left endpoint: C = the critical point: C = the right endpoint: C = Based on the above, we may conclude that the distance from B to C that minimizes the cost is m
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