How Should Cable Be Laid to Minimize Costs Across a River?

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SUMMARY

The discussion focuses on minimizing the cost of laying an underground telephone cable between two boathouses located 600 meters apart along the riverbanks, with the river width being 200 meters. The cost of laying cable is $150 per meter underwater and $80 per meter on land. To minimize costs, one must derive an expression for the total length of the cable based on the chosen landing point on the opposite bank. The optimal solution involves calculating the cost for different cable routing scenarios, including direct crossing and combinations of underwater and land routes.

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  • Understanding of basic geometry, specifically the Pythagorean theorem.
  • Familiarity with algebraic expressions and cost functions.
  • Knowledge of optimization techniques in mathematics.
  • Ability to visualize and diagram geometric problems.
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  • Learn how to derive cost functions based on geometric parameters.
  • Study optimization techniques, particularly in calculus.
  • Explore practical applications of the Pythagorean theorem in real-world scenarios.
  • Investigate cost minimization strategies in engineering projects.
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Mathematicians, engineering students, project managers in telecommunications, and anyone involved in cost analysis for infrastructure projects.

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An underground telephone cable is to be laid btwn 2 boathourses on opposite banks of a straight river. One boathouse is 600 meters downstream from the other. The river is 200 meters wide. If the cost of laying cable is $150 per metere underwater and $80 per metere on land, how should the cable be laid to minimize cost?

Can someone help me figure this out and show how they did it?

Thank u so much if u can.
 
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shreena03 said:
An underground telephone cable is to be laid btwn 2 boathourses on opposite banks of a straight river. One boathouse is 600 meters downstream from the other. The river is 200 meters wide. (See Figure 35). If the cost of laying cable is $150 per metere underwater and $80 per metere on land, how should the cable be laid to minimize cost?

Can someone help me figure this out and show how they did it?

Thank u so much if u can.


It is not the policy of this site to simply do homework for people. You will not be able to come here to request a solution during your test.

What you need to do is find an expression for the total length of the wire. Consider that you must cross the river, so start by drawing yourself a picture of the situation. Can you find an expression for the distance across the river in terms of the distance along the shore. You have all the information required to do this.

Show us what you are thinking.
 
Think of it this way.

Draw a diagram. Think of the possibilities. There are really three. Do you take the hypotenuse route straight to it? Or do you go directly across the water, then along the opposite side?

Or perhaps you land somewhere between and carry on the rest from that point?

Find an equation. What are you looking for? The cost of it. What do you want to do? Minimize or maximize?
 

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