Annual profit in thousands of dollars is given by the function, P(x) = 5000 - (1

Question

Annual profit in thousands of dollars is given by the function, P(x) = 5000 - (1000/(x-1)), where x is the number of items sold in thousands, x > 1.
1.describe the meaning of the number 5000 in the formula
2.describe the meaning of the number 1 in the formula
3.find the profit for 5 different values of x
4.graph the profit function over its given domain; use the 5 values calculated in part 3 to construct the graph and connect these points with a smooth curve in Excel or another graphing utility. Insert the graph in a Word file and attach the graph in a Word file to the class DB thread.
5.will this profit function have a maximum, if so, what is it?
6.what steps should the company take to prepare for your answer to part 5?

Explanation / Answer

1) The 5000 is the initial condition of the function, that is, it modifies the function such that the final values are measured by how far they are from 5000 rather than 0. For this specific equation, it is also the upper limit of the possible profit.

2) Assuming the 1 they are referring to is the 1 in "x>1". The 1 is the boundry of the function, that is, the model is only true when more than 1000 units are being sold.

3) Profit for x=2:
P(2)=5000-1000/(2-1)=5000-500=4500
Profit for x=5
P(5)=5000-1000/(5-1)=5000-250=4750
Profit for x=11
P(11)=5000-1000/(11-1)=5000-100=4900

4)Using the above points the graph will look something like:
http://www.wolframalpha.com/input/?i=PLOT%285000+-+%281000%2F%28x-1%29%29%2Cx%3D1%2Cx%3D20%29

5)Yes, the maximum is at P=5000. This can be proven explicitly using calculus(msg me for the full proof). Consider that as x gets arbitrarily large, the second term gets arbitrarily small, thus at some giant number say x=10000000, the function becomes:
P=5000-1000/(99999999)5000-0=5000.

6)It should come up with a new manufacturing process with a better profit margin for very large x (x>>1). This would be based on when the company was no longer making a signifigant amount more for producing a very large number of items, say x>50. (@ x=50, the profit margin is about 4950, at x=100, the profit margin is only 4990..a gain of only $40 even though 50000 more units are being sold)

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