Quadratic Optimization

[jess5085]

Q. A manufacturer of pot-bellied stoves has the following situation to consider:

If x units are made per week, each one will cost \(\displaystyle \, 50\, +\, \frac{400}{x}\,\) dollars and the total receipts per week for selling with be \(\displaystyle \, 550x\, -\, 2x^2\,\) dollars.

How many pot-bellied stoves should be made per week in order to maximize profits?

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I'm not sure if I'm attempting it right.
So far I've got:
Total revenue = (550x-2x^2) - (50-400/x)
=550x-2x^2-50-400/x

=-2x^2......

Im not sure what to do after this.

Thanks for your help!

 

[stapel]

Q. A manufacturer of pot-bellied stoves has the following situation to consider:

If x units are made per week, each one will cost \(\displaystyle \, 50\, +\, \frac{400}{x}\,\) dollars and the total receipts per week for selling with be \(\displaystyle \, 550x\, -\, 2x^2\,\) dollars.

How many pot-bellied stoves should be made per week in order to maximize profits?

---

I'm not sure if I'm attempting it right.
So far I've got:
Total revenue = (550x-2x^2) - (50-400/x)

The "minus" sign in front of the 400/x should be a "plus" sign.

=550x-2x^2-50-400/x

The sign is magically corrected here, but make sure to correct it in the previous line, too!

=-2x^2......

Im not sure what to do after this.

Start by thinking this out a bit more carefully. So far, you are including the cost of only one unit. Instead, you need to include the cost of all x units!

 

[Steven G]

As Stapel says: 550x - 2x^2 - x(50 + 400/x) = profit

Distribute the x and then combine like terms. The max value for Profit will be when x= -b/(2a). The a and b come from Profit = ax^2 + bx +c

 

[stapel]

As Stapel says: 550x - 2x^2 - x(50 + 400/x) = profit

If you are not familiar with derivatives, then you are not ready for this problem.

By correcting the "cost" expression, the variable in the denominator clears out, and regular quadratic / algebraic methods can be applied. No derivatives necessary.

 

[jess5085]

Thanks for your help. I worked it out.