How Many Items Must Be Sold for a Company to Break Even?

[AGA19]

Homework Statement

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.

Homework Equations

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

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!

 

[Ultramilk]

1620sqrt(x)/x - formula for how much each item is worth when sold
40x - the cost of every item produced
12200 - overhead

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.

 

[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.

 

[AGA19]

Can anybody further explain this please?

 

[HallsofIvy]

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.