curve sketching with polynomial functions

[john3j]

Ok, here is the problem, but I am not asking for someone to do the work for me. I am instead asking for someone to explain the steps so that I can complete it on my own.

The productivity rating of an individual worker at the Cruz Corporation assembley line is based on the number of tasks accomplished, mistakes made, and responsiveness to difficulties encountered. The average of all scores allows the company to use a simple model based on time on the floor given by PR=-0.4x^3+2x^2+10x+5, where x is in hours at work. A PR score of 20 is acceptable and a score of 20 is acceptable and a score of 40 is highly unusual.

When are workers' scores the highest?
Design an 8-hour day where workers do the most demanding jobs for about 6 hours and have 2 hours for less stressful work. Explain your reasoning.

Any help would be appreciated, but steps to figure it out on my own would be most helpful.

Thank you!

 

[JeffM]

Ok, here is the problem, but I am not asking for someone to do the work for me. I am instead asking for someone to explain the steps so that I can complete it on my own.

The productivity rating of an individual worker at the Cruz Corporation assembley line is based on the number of tasks accomplished, mistakes made, and responsiveness to difficulties encountered. The average of all scores allows the company to use a simple model based on time on the floor given by PR=-0.4x^3+2x^2+10x+5, where x is in hours at work. A PR score of 20 is acceptable and a score of 20 is acceptable and a score of 40 is highly unusual.

When are workers' scores the highest?
Design an 8-hour day where workers do the most demanding jobs for about 6 hours and have 2 hours for less stressful work. Explain your reasoning.

Any help would be appreciated, but steps to figure it out on my own would be most helpful.

Thank you!

John

I can't help you because the problem is obscure to me. Why is 20 repeated twice? What relevance do stressful and non-stressful hours have when the model is simply in terms of hours worked without reference to stress. Why graph? Why not take the derivative? What exactly does the problem say?

 

[john3j]

John

I can't help you because the problem is obscure to me. Why is 20 repeated twice? What relevance do stressful and non-stressful hours have when the model is simply in terms of hours worked without reference to stress. Why graph? Why not take the derivative? What exactly does the problem say?

Jeff,

I am sorry that my brain wasnt processing what was in front of me last night. With everything going on at home and at work, its a wonder I can even focus on Calculus. Anyways, the problem is number 44 and is in the picture. Please provide any assistance or instructions?



I appreciate the time you take to help me!

Thanks,
John

 

[JeffM]

Jeff,

I am sorry that my brain wasnt processing what was in front of me last night. With everything going on at home and at work, its a wonder I can even focus on Calculus. Anyways, the problem is number 44 and is in the picture. Please provide any assistance or instructions?

View attachment 2523

I appreciate the time you take to help me!

Thanks,
John

Let's take this in steps. Question 44a does not require any curve sketching. You are given an explicit function relating hours worked and productivity. You are asked where that function is maximized. Find where the first derivative is zero and see whether it is a maximum. Implicitly, this function is bounded: no one can work for less than 0 hours or more than 24 hours in a day. In a different thread, I explained to you about needing to examine boundary conditions on bounded functions. Nevertheless, this is all straight calculus. A graph may help you visualize what is going on, but it is not at all necessary.

Question 44b is entirely different. Some may not like it because it is only partially a math question. That's why I do like it. It is asking you to look at a graph and, subject to certain constraints, apply some common sense to it. Suppose you do the graph and find that productivity actually falls during the first hour of work. People have hangovers or are still thinking about their morning flirtation in the cafeteria. In that case, it might make sense to schedule undemanding work for the first hour of the shift until people get in the groove. So what this problem is asking you to do is to sketch the graph so you can look at it as a whole and think about what it means and how you can use that information to make an improvement.

 

[john3j]

Thank you very much Jeff. I have completed this and believe that I understand. Thanks for not insulting me like someone did on another post of mine.

Let's take this in steps. Question 44a does not require any curve sketching. You are given an explicit function relating hours worked and productivity. You are asked where that function is maximized. Find where the first derivative is zero and see whether it is a maximum. Implicitly, this function is bounded: no one can work for less than 0 hours or more than 24 hours in a day. In a different thread, I explained to you about needing to examine boundary conditions on bounded functions. Nevertheless, this is all straight calculus. A graph may help you visualize what is going on, but it is not at all necessary.

Question 44b is entirely different. Some may not like it because it is only partially a math question. That's why I do like it. It is asking you to look at a graph and, subject to certain constraints, apply some common sense to it. Suppose you do the graph and find that productivity actually falls during the first hour of work. People have hangovers or are still thinking about their morning flirtation in the cafeteria. In that case, it might make sense to schedule undemanding work for the first hour of the shift until people get in the groove. So what this problem is asking you to do is to sketch the graph so you can look at it as a whole and think about what it means and how you can use that information to make an improvement.