Optimisation problem: Assume the consumption of coal by a certain steamer is represen

[Rorite]

Hi, I need help with the following problem, can't seem to figure it out. Assume the consumption of coal by a certain steamer is represented by the formula y =0.3+0.001v^3 ,where y is the number of tons of coal burned per hour and v is the speed expressed in nautical miles per hour. The cost of wages, financing, and depreciation of that ship are together equal, per hour, to the cost of 1 ton of coal. What speed will make the total cost of a voyage of 1000 nautical miles a minimum? And, if coal costs $100 per ton, what will that minimum cost of the voyage amount to?

Answers are:
Speed: 8.66
Time taken: 115.47
Minimum cost: 22.5k

 

[mmm4444bot]

Assume the consumption of coal by a certain steamer is represented by the formula y =0.3+0.001v^3 ,where y is the number of tons of coal burned per hour and v is the speed expressed in nautical miles per hour. The cost of wages, financing, and depreciation of that ship are together equal, per hour, to the cost of 1 ton of coal. What speed will make the total cost of a voyage of 1000 nautical miles a minimum? And, if coal costs $100 per ton, what will that minimum cost of the voyage amount to?

You agreed to follow the guidelines, when you joined this site. Have you read the guidelines, yet?

What have you thought about so far? What did you try? At what point did you get stuck? Is there something specific that's confusing you?

Please provide this missing information. Thank you! :cool:

 

[HallsofIvy]

Hi, I need help with the following problem, can't seem to figure it out. Assume the consumption of coal by a certain steamer is represented by the formula y =0.3+0.001v^3 ,where y is the number of tons of coal burned per hour and v is the speed expressed in nautical miles per hour. The cost of wages, financing, and depreciation of that ship are together equal, per hour, to the cost of 1 ton of coal.

So, since coal costs $100 per ton, how much does it cost tur run the ship, other than coal?
At speed v, in nautical miles per hour, how long will it take to go 1000 nautical miles? So how much will everything except coal cost for the trip? How much coal will it take to go 1000 nautical miles at that speed?

What speed will make the total cost of a voyage of 1000 nautical miles a minimum? And, if coal costs $100 per ton, what will that minimum cost of the voyage amount to?

Answers are:
Speed: 8.66
Time taken: 115.47
Minimum cost: 22.5k

If I were your teacher, I would NOT accept these answers! The correct speed is NOT "8.66", it is "8.66 nautical miles per hour". The correct time taken is NOT "115.47", it is "115.47 hours". The correct minimum cost is NOT "22.5k", it is "22.5k dollars".

 

[Rorite]

Stuck

You agreed to follow the guidelines, when you joined this site. Have you read the guidelines, yet?

What have you thought about so far? What did you try? At what point did you get stuck? Is there something specific that's confusing you?

Please provide this missing information. Thank you! :cool:

Basically I do not know where to go from here, I basically know I find the derivate with respect to the speed v.

 

[Rorite]

My mistake forgot to provide the answers' units. The problem I am facing is, I just do not know how to tackle the problem, I know the procedure for optimization but I am having a hard time formulating the function I need to differentiate. Can anyone give me a hint on where to start atleast?

 

[HallsofIvy]

I thought I did. At speed v, in km/h, how many hours will it take to go 1000 nm? You are told that the consumption of coal, per hour, at speed v, is given by 0.3+0.001v^3. So how many tons of coal will be required for that many hours? Since coal cost $100 per ton, how much will that coal cost? There are also additional costs ("wages, financing, and depreciation of that ship") of $100 per hour. So what is the total cost of the trip? Those will all be functions of v, of course.