# Calculus Word Problem: Profit

1. Oct 16, 2011

### AGA19

1. The problem statement, all variables and given/known data

A company sells a certain product and determines that the best price that each item/product must be sold for is given by the formula P=1,620 Square Root X Over X. AKA P=1,620 Square Root X divided by X. X is the # of items sold. Each item cost's $40 to produce4. All items produced are sold and the overhead cost is$12,200. Find the # of items that must be sold for the company to break even.

2. Relevant equations

12,200=1,620 Square root X/X???

3. The attempt at a solution

Have no idea what I'm doing here. I'm guessing that you plug 12,200 in for P. But then how do you find X? I'm so lost. Please help!

2. Oct 16, 2011

### Ultramilk

1620sqrt(x)/x - formula for how much each item is worth when sold
40x - the cost of every item produced

p(x) = 1620sqrt(x)/x - 40x - 12200
1620sqrt(x)/x should be positive as you're making money, and the other two should be negative because you're spending it to start up.

So now find d/dx p(x).

edit: Actually, if it's just to find when the company break even and not an optimize profit, set p(x) = 0 and solve from there.

3. Oct 17, 2011

### AGA19

So the problem should now look like. 0=1620 square root X/X -40X -12200. So I'd add 12,200 to both sides.

12,200+40 X=1,620 square root X/X???

Sorry Calculus is killing me this semester.

4. Oct 17, 2011

### AGA19

Can anybody further explain this please?

5. Oct 17, 2011

### HallsofIvy

Staff Emeritus
You have
$$P= \frac{1620\sqrt{x}}{x}$$
but I am puzzled as to why that is not written as
$$P= \frac{1620}{\sqrt{x}}$$
or just $P= 1620x^{-1/2}$

And, indeed, I serious problem with that! It says that the more you raise the price, the more you sell! In order to sell 4 times as much (replace x with 4x) your prices should be twice as much. Normally, raising the price of something discourages people from buying and reduces the amount sold.

Are you sure it wasn't
$$P(x)= \frac{1620- \sqrt{x}}{x}$$
or something like that?

Anyway, taking the formula you gave for price, the income from selling x things is just the price times x:
income= $1620\sqrt{x}$

You say they each cost $40 to make and there are overhead costs of$12200. So making "x" of them costs 40x+ 12200

"Break even" is when income equals cost:
$$1620\sqrt{x}= 40x+ 12200$$

There is no "profit" and you don't need calculus to solve that- just square both sides to get a quadratic equation and then solve that equation.