# What Speed Minimizes Total Cost for a 1000-km Truck Journey?

• fghtffyrdmns

## Homework Equations

$$C(v) = \frac{v}{100}+\frac{25}{v}$$

## The Attempt at a Solution

$$C(v) = \frac{v^2+2500}{100v}$$

I simplified it into one expression. From here, I differentiate and find the minimum speed. I divide $40/h by the speed to get$/km which I can then use to solve again?

I do not think this is right.

fghtffyrdmns said:

The cost of fuel per kilometre for a truck traveling $$v$$ kilometres per hour is given by the equation $$C(v) = \frac{v}{100}+\frac{25}{v}$$. Assume the driver is paid $40/h. What speed would give the lowest cost, including fuel and wagesm for a 1000-km trip? ## Homework Equations $$C(v) = \frac{v}{100}+\frac{25}{v}$$ ## The Attempt at a Solution $$C(v) = \frac{v^2+2500}{100v}$$ I simplified it into one expression. From here, I differentiate and find the minimum speed. You don't want the minimum speed, you want the minimum cost. fghtffyrdmns said: I divide$40/h by the speed to get $/km which I can then use to solve again? I do not think this is right. I agree. The total cost for the trip is the cost for fuel plus the driver's wages. You are given the cost per km as a function of the speed, but you also need to add in the driver's wages, which are also a function of the speed (the faster he drives, the fewer hours, so the less he gets). That is the part I am stuck at. If I solve the derivative, I would find out the speed which would give the lowest fuel cost per kilometre. While I tried to do this, I got a local max at x=-50 and local min at x=50. Something does not seem right. fghtffyrdmns said: While I tried to do this, I got a local max at v=-50 and local min at v=50. Something does not seem right. If I am trying to find the minimum cost, I should be using maximum speed which occurs at v=-50. fghtffyrdmns said: That is the part I am stuck at. If I solve the derivative, I would find out the speed which would give the lowest fuel cost per kilometre. What is the function you have for the total cost of the trip? What is it that you are taking the derivative of? You DO NOT want to take the derivative of C(v). Mark44 said: What is the function you have for the total cost of the trip? What is it that you are taking the derivative of? You DO NOT want to take the derivative of C(v). Would I have to multiply the$40/h into C(v) to get the total cost?

Edit: I know I have to do something with 40.

Last edited:
Yes, you have to do something with the $40/hr. Look at what I wrote at the bottom of post #2. If what I have isn't clear, ask questions about it. Mark44 said: Yes, you have to do something with the$40/hr.

Look at what I wrote at the bottom of post #2. If what I have isn't clear, ask questions about it.

I think I understand this now.

You know that speed is equal to distance/time. Time = distance/speed.

$$t = \frac{1000}{v}$$

Now, to make the total cost function you would write: $$C(t) = (\frac{v}{100}+\frac{25}{v})(1000) + 40t$$

Substitute d/t into the equation, differentiate and solve, you would get how many hours (minimum). Then you can find speed.

That's a little different from what I did, but it seems reasonable, so I think it will work. Notice that your cost function represents fuel cost for the trip + cost of driver's wages, which is what I've been saying you need to work with.

## 1. How can I optimize speed and cost in my experiment?

To optimize speed and cost in your experiment, you can start by carefully planning and designing your experimental procedure. This includes choosing the right materials and equipment, as well as minimizing unnecessary steps. Additionally, consider automating certain parts of your experiment to increase efficiency and reduce costs. Finally, regularly analyze and evaluate your data to make necessary adjustments and improvements.

## 2. What impact does optimizing speed and cost have on the accuracy of my results?

Optimizing speed and cost can actually improve the accuracy of your results. By reducing the time and resources needed for your experiment, you can minimize potential errors or variables that could affect your results. It also allows for quicker feedback and adjustments, leading to more accurate and reliable data.

## 3. Are there any specific techniques or tools that can help with optimizing speed and cost?

Yes, there are various techniques and tools that can aid in optimizing speed and cost. This includes using statistical methods to design efficient experiments, utilizing computer simulations, and implementing cost-saving measures such as group testing or using reusable materials. Collaborating with other researchers and staying updated on new technologies can also help improve speed and reduce costs.

## 4. What are the potential drawbacks of optimizing speed and cost in scientific research?

While optimizing speed and cost can bring numerous benefits, there are also potential drawbacks to consider. For example, cutting corners to save time or money could compromise the quality and reliability of your results. It could also limit the scope and depth of your research. Therefore, it is important to strike a balance between speed and cost optimization and maintaining scientific rigor.

## 5. How can I justify the cost of optimizing speed in my research proposal?

When writing a research proposal, it is important to clearly explain the potential benefits of optimizing speed, such as reducing the overall cost or increasing the efficiency of the project. Additionally, you can provide evidence of similar studies that have successfully optimized speed and cost, and how it has improved the quality of their results. You can also highlight any potential cost-saving measures or collaborations that will help justify the cost of optimizing speed in your research.

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