Calculus minimum

  • #1
The cost of running a ship at a constant spped of vkm/h is 160 + (v^3)/ 100 dollars per hour.

a) Find the most economical speed for the journey, and the minimum cost.
b) If the ship were to have maximum speed of 16km/h, find what the minimum cost would be.
 

Answers and Replies

  • #2
CompuChip
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The cost is given as a function
[tex]C(v) = 160 + v^3 / 100[/tex]
How do you find the minimum of a function?

Then what do you think you should do to get b?
 
  • #3
The function you present has no minimum value due to the fact that it is cubic with no other terms. You can use calculus as follows to prove this:

C(v) = 160 + .01 * v^3
C’(v) = .03 * v^2
0 = .03 * v^2
v^2 = 0 / .03 = 0
v = ±0 = 0

The function C'(v) is the derivative of the cost function. By setting this to zero, you are solving for the extrema of the function C(v). Extrema include minimums (what you are looking for), maximums, and points of inflection. In this case, there is a point of inflection at velocity = 0 km/hour. You can check this by either graphing the function or checking the second derivative to figure out which type of extrema the point is. You should notice however that functions of the form v^3 always have a point of inflection at v=0 with no local maximums or minimums to be found.

To answer your question however based on the function, because the cost per hour gets exponentially larger the faster the ship travels, the slower the ship goes, the more cost efficient the journey will be. In fact, as the ships speed approaches zero, the cost of running the ship approaches $160 per hour. This however is assuming the cost is independent of the distance to be traveled...

...however, from reading part (a) and part (b) of your question, I assume you can use other variables to express your answer, and in this case, the distance of the journey is crucial in solving your problem (distance will be 'd'):

C(v) = 160 + .01 * v^3
t(v) = d / v

The second function there is the length of the journey given d distance and v velocity. By multiplying these functions together, you get the total cost of a journey in terms a constant (given) distance and an unknown velocity. Taking the derivative of this function then allows you to solve for the minimum cost of that journey:

C(v)*t(v) = (160 + .01 * v^3) * (d / v)
cost(v) = 160*d*v^-1 + .01*d*v^2

Knowing the distance to be a given constant you can solve for a minimum cost with velocity 'v' using the above equation. As part (b) to your question alludes, the faster the ship goes, the more cost effective it will be up until a point. In the case of 16 km/hr being the maximum speed of the ship, the following would be the minimum cost of the journey in terms of 'd' distance:

cost(v) = 160*d*v^-1 + .01*d*v^2
cost(v) = 160*d*(16)^-1 + .01*d*(16)^2
cost(v) = 10*d + 2.56*d
cost(v) = 12.56*d

In this case, the cost of running the ship is $12.56 per hour if the ship goes at its maximum speed throughout the journey. Ultimately though, the pitfall to this kind of question is over-analyzing the question using math. Compuchip's response is 'correct' in a sense of the word, but doesn't truly answer the question being asked, and the question is fairly vague and sounds like it's missing a part considering it asks about a 'journey' in part (a) but doesn't given any facts about the journey (ie distance).
 
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  • #4
534
1
The problem is probably really asking this: Find the speed that will minimize the cost over some fixed distance. If the trip takes time t, the total cost is C(v)t. If you travel a fixed distance d, the time is then d/v, so the cost is C(v) d/v. In that case, you want to minimize C(v)/v = 160/v + v2/100.


Answer:
v = 20
 
Last edited:
  • #5
That sounds about right; the distance should be kept in their though just in case:

C(v)*d/v = d*160/v + d*v^2/100

But yes, you can easily prove that the distance is negligible, meaning that the optimal velocity is the same for all distances.
 
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