# Optimization problem

• Dec 13th 2012, 03:57 AM
PhizKid
Optimization problem
The cost of manufacturing x number of pencils is given by: C = x^2 + 3x + 13
If each pencil is sold for $5, maximize the profit. I thought profit was equal to the total amount made and subtract the cost of making them, so: profit = (5x) - (x^2 + 3x + 13) and then I differentiated and set it equal to 0 and found that x = 1 is the best to maximize the profit, but it's wrong. I think I set up the profit equation incorrectly. How do I do this? • Dec 13th 2012, 07:49 AM HallsofIvy Re: Optimization problem Quote: Originally Posted by PhizKid The cost of manufacturing x number of pencils is given by: C = x^2 + 3x + 13 If each pencil is sold for$5, maximize the profit.

I thought profit was equal to the total amount made and subtract the cost of making them, so: profit = (5x) - (x^2 + 3x + 13) and then I differentiated and set it equal to 0 and found that x = 1 is the best to maximize the profit, but it's wrong. I think I set up the profit equation incorrectly. How do I do this?

What, exactly, is the question? "Maximise the profit" is vague. You are correct that the maximum profit is when x= 1. Perhaps they want you state the actual profit in that case?
• Dec 13th 2012, 11:27 AM
hollywood
Re: Optimization problem
There is a possible twist to this problem. You have correctly calculated:

$P = (5x) - (x^2 + 3x + 13)=-x^2+2x-13$

$\frac{dP}{dx}=-2x+2$

so x=1 is the maximum. But notice that P=-12 at this maximum. Surely it would be possible to close the pencil-making operation completely and thereby achieve a greater profit (or smaller loss, if you prefer) with P=0. Though, like HallsofIvy said, the question is vague. Perhaps they just want you to say that it's not possible to make a profit.

- Hollywood