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.