The profit (P), in dollars, for a company is modeled by the funtion P=-750x^2+15,000x, where x is the number of items produced. For which values of x will the company losr money?

you want where P < 0

P(x) = 750x(20-x)
You know that P=0 when x=0 or 20

Since the graph is a parabola opening downward, P>0 between the roots.

So, P<0 if x>20

To determine for which values of x the company will lose money, we need to find the values of x where the profit (P) is negative.

The profit function is given by P = -750x^2 + 15,000x.

To find the values of x for which the company will lose money, we need to find the values where P < 0.

So we can set up the inequality:

-750x^2 + 15,000x < 0

Now, let's solve this inequality.

First, let's factor out common terms:

x(-750x + 15,000) < 0

Now, we have two factors: x and -750x + 15,000.

To determine the sign of each factor, we set them both equal to zero:

x = 0 and -750x + 15,000 = 0

Solving the equation -750x + 15,000 = 0, we find:

-750x = -15,000

Dividing both sides by -750, we get:

x = 20

Therefore, we have two critical points: x = 0 and x = 20.

Now, we can use these critical points to divide the number line into three intervals.

For x < 0, both factors are negative, and the product is positive. So, this interval does not satisfy the inequality.

For 0 < x < 20, x is positive and -750x + 15,000 is positive, so the product is negative. This is the interval where the company will lose money.

For x > 20, x is positive, but -750x + 15,000 is negative, so the product is positive. This interval does not satisfy the inequality.

Therefore, the company will lose money when 0 < x < 20.