Word Problem - Functions

[N0ximis]

Need help with the following word problem:

You are going to sell oranges by the box full. The first supplier will sell you a box of 20 oranges for $16.00 per package, but there is an additional charge of $300.00 per month to purchase oranges from the company.


1. Find and show you how you found the formula for C(N), which is what it costs your group for oranges, in which n is the # of boxes of oranges.

For this problem, I came up with the formula: C(N) = ($16.00 N)+ $300 Am I on the right track?

2. Determine P(p), which models your profit if you buy your oranges from this company. Do not forget to rewrite your cost function in terms of your selling price, p.


3. What kind of function is P.


4. When you charge this price, how many boxes are sold?


5. Find the maximum profit.




I need the solutions and steps to derive the solutions to these five questions.
I sort of know how to get the answers, but I just want to make sure my conclusions are valid. All help is appreciated.

 

[mmm4444bot]

For [the first part of] this problem, I came up with the formula: C(N) = ($16.00 N)+ $300 Am I on the right track?

Yes, your cost function is correct.

Let's stick with "n" as the variable name, however, because "N" and "n" are different names. Also, we do not write dollar signs in equations; it's understood from "cost" that the three numbers C(n), 16n, and 300 are all dollar amounts.

C(n) = 16n + 300

I need the solutions and steps to derive the solutions to these five questions. I sort of know how to get the answers, but I just want to make sure my conclusions are valid.

The statement in red is kinda funky. If we give you the steps for deriving the solutions, and we give you the solutions, then the assignment becomes mostly moot.

If you desire to check the validity of your conclusions, then please post them. Or, you may show any work or reasoning that you've done. We'll check these for you. Otherwise, please ask specific questions about the parts where you get stuck. Thank you.

 

[N0ximis]

Yes, your cost function is correct.

Let's stick with "n" as the variable name, however, because "N" and "n" are different names. Also, we do not write dollar signs in equations; it's understood from "cost" that the three numbers C(n), 16n, and 300 are all dollar amounts.

C(n) = 16n + 300





The statement in red is kinda funky. If we give you the steps for deriving the solutions, and we give you the solutions, then the assignment becomes mostly moot.

If you desire to check the validity of your conclusions, then please post them. Or, you may show any work or reasoning that you've done. We'll check these for you. Otherwise, please ask specific questions about the parts where you get stuck. Thank you.

Okay. For question 2, since income - cost is net profit, this is what I came up for P(p):

P(n) = pn - 16n - 300 =-(p-16)n - 300.

Because Income = pxn right? I know the function type is based on the power. If the power is 1, then the function is linear. Since the power is 1, the function is linear.

I could not find an answer to question 4. Price certainly does correlate with sales. However, determining a solid and exact price is impossible.

For last question, I concluded that the max is positive infinity because there is no limit to how many oranges you can sell. But I also think it could be undetermined because we don't know what the maximum is. I'm not sure what to say, positive infinity or cannot be determined.

 

[JeffM]

Okay. For question 2, since income - cost is net profit, this is what I came up for P(p):

P(n) = pn - 16n - 300 =-(p-16)n - 300.

Because Income = pxn right? I know the function type is based on the power. If the power is 1, then the function is linear. Since the power is 1, the function is linear.

Do not use x to represent multiplication in algebra. I kept scratching my head wondering what variable was represented by x. Use * for multiplication.

I could not find an answer to question 4. Price certainly does correlate with sales. However, determining a solid and exact price is impossible.

For last question, I concluded that the max is positive infinity because there is no limit to how many oranges you can sell. But I also think it could be undetermined because we don't know what the maximum is. I'm not sure what to say, positive infinity or cannot be determined.

It appears likely that part of the statement of the problem is missing. Usually there is some sort of function that relates number sold to price.

That is, there is usually something that looks like n = S(p). So you are correct that p times number sold = revenue and that profit equals revenue minus cost, but you cannot say what kind of equation it is unless you know what S(p) is. Furthermore, your cost function is going to be dependent on S(p) also. The statement of the problem specifically warns you to "rewrite" your cost function.

I am curious. Is this problem from a high school algebra course or from a beginning economics course? It's a great problem for an econ course, but I am not sure that it does not expect more knowledge of economic models than would have been common back when I was studying algebra.

You may be able to find a finite maximizing price and quantity once you know what the full model is. Obviously a model that tells you to sell an infinite number of anything is either invalid generally or is valid only within a certain domain.