Calculate Cost for Most Economical Speed & Max Speed for Ship Voyage

In summary, the optimal velocity for the journey is 10 km/h, with a minimum cost of $12.56 per hour if the ship were to have a maximum speed of 16 km/h. The cost function C(v) is a cubic function with no minimum value, but by setting the derivative to zero, we can find the extrema of the function. However, in this case, there is a point of inflection at velocity = 0 km/h, meaning there are no local maximums or minimums to be found. The cost of running the ship is exponentially larger the faster the ship travels, making slower speeds more cost efficient. The distance of the journey is crucial in solving for the minimum cost, as shown in the
  • #1
disfused_3289
13
0
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.
 
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  • #2
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
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
 
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  • #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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1. What is the purpose of calculating the cost for the most economical speed and maximum speed for a ship voyage?

The purpose of this calculation is to determine the most efficient speed for a ship to travel in order to minimize costs and maximize profits during a voyage. By considering factors such as fuel consumption, time, and potential delays, this calculation can help optimize the overall performance of a ship.

2. How is the cost for the most economical speed and maximum speed for a ship voyage calculated?

The cost for the most economical speed and maximum speed for a ship voyage is calculated by taking into account various factors such as the distance of the voyage, fuel consumption rates at different speeds, and potential delays. This calculation is typically done using mathematical models and simulations.

3. What factors are considered when determining the most economical speed for a ship voyage?

Factors that are typically considered when determining the most economical speed for a ship voyage include the distance of the voyage, fuel prices, fuel consumption rates at different speeds, potential delays due to weather or traffic, and the cost of labor and maintenance. Other factors may also be taken into account depending on the specific voyage and circumstances.

4. Why is it important to calculate the most economical speed for a ship voyage?

Calculating the most economical speed for a ship voyage is important because it can help shipping companies and operators make informed decisions about the performance of their ships. By determining the most efficient speed for a voyage, companies can save on fuel costs and potentially reduce the overall cost of the voyage. This calculation can also help identify potential issues or delays that may impact the profitability of the voyage.

5. How does the maximum speed for a ship voyage impact the overall cost?

The maximum speed for a ship voyage can impact the overall cost in several ways. Firstly, traveling at higher speeds typically results in higher fuel consumption and therefore, higher costs. Additionally, higher speeds may also increase the risk of potential delays or accidents, which can result in additional costs for repairs or lost time. Therefore, it is important to balance the need for speed with the cost implications when planning a ship voyage.

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