Optimization problem that makes no sense

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SUMMARY

The optimization problem involves minimizing the cost per mile traveled by a ferry boat, where fuel costs are proportional to the cube of the speed. The ferry consumes $100 of fuel per hour at 10 miles per hour, with additional operational costs of $675 per hour. The correct speed to minimize costs is determined to be 15 miles per hour, aligning with the solution provided in the textbook. The key takeaway is understanding the relationship between speed and fuel costs, which is cubic in nature.

PREREQUISITES
  • Understanding of cubic functions and their properties
  • Basic knowledge of optimization techniques in calculus
  • Familiarity with cost analysis in operational contexts
  • Ability to set up and solve equations based on proportional relationships
NEXT STEPS
  • Study cubic functions and their applications in real-world scenarios
  • Learn about optimization methods in calculus, specifically for minimizing costs
  • Explore cost analysis techniques for transportation and logistics
  • Practice setting up and solving proportional equations in various contexts
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Students in mathematics or engineering fields, transportation analysts, and anyone involved in cost optimization for operational efficiency will benefit from this discussion.

Burjam
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Homework Statement



The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the cost of running this ferry (labor, maintenance, and so on) is $675 per hour. At what speed should it travel so as to minimize the cost per mile traveled?

Homework Equations



N/A

The Attempt at a Solution



I'm having trouble setting up functions to solve this problem. I could do it if I just had some solid functions to work with. The thing that's really getting me is the wording.

The problem states:

The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour.

Last time I checked, 10^3 is 1000. What's going on here?
 
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Burjam said:

Homework Statement



The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the cost of running this ferry (labor, maintenance, and so on) is $675 per hour. At what speed should it travel so as to minimize the cost per mile traveled?

Homework Equations



N/A

The Attempt at a Solution



I'm having trouble setting up functions to solve this problem. I could do it if I just had some solid functions to work with. The thing that's really getting me is the wording.

The problem states:
The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour.


Last time I checked, 10^3 is 1000. What's going on here?
It doesn't say that the cost is the cube of the speed, just that the cost is proportional to the cube of the speed. For example, if the boat's speed was increased to 20 mph, it would use $800 of fuel per hour.
 
Ok did it. I got 15mph which is what the back of the book says. Thanks
 

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