Maximizing Profit: Balancing Cost and Price for 300 Units

[atkinslm]

Homework Statement

A company can do 300 units per week at a weekly cost per unit of $58 for labor and supplies. If the charge is $100 per unit, then it would have clients for 300 units. However, for each $5 price increase, it could expect to have 10 fewer units.

Homework Equations

The Attempt at a Solution

 

[JonF]

Can you come up with a formula for profit?

Let's do the easy one first, when the company doesn't charge anything extra:
P_i(x) = (number of units)(sell price - cost) = 300(100-58)
Fairly straight forward, now let's modify this with a variable “x” which will represent how many times they will charge an extra $5.

P(x) = (number of units)(sell price - cost) = (300-10x)(100+5x-58) = (300-10x)(42+5x)
Now what we do from here depends on what class you are in. If you are in an algebra course, you should recognize this as a parabola and know how to graph it. If you are in calculus I would expand it, take the derivative, and set it equal to zero.

 

[atkinslm]

Thanks for your help so far. I am in Precalculus. I need to plug these formulas into an excel spreadsheet.

 

[atkinslm]

I have the formulas in the spread sheet. I think all is going well. I need to know what type of rule is this. I would really like to be able to figure this out. Is it the product of two polynomials?

 

[JonF]

It is two polynomials multiplied. You should be able to do this by hand by using the FOIL method then putting it standard form for a parabola. Picture a parabola, what point would be the max or min?

 

[atkinslm]

Thanks a lot!