The profit of a company producing x thousand units is given by P(x)=-2x^2+38x-120
Find the values of x for which the company makes a profit. Explain and justify your answer.
[Hint: The company makes a profit when P(x)>0]
To find the values of x for which the company makes a profit, we need to find the range of x where P(x) > 0. We can start by factoring the quadratic equation to find the critical points (where P(x) = 0).
P(x) = -2x^2 + 38x - 120
We can factor out a -2:
P(x) = -2(x^2 - 19x + 60)
Now we try to factor the quadratic expression inside the parentheses:
x^2 - 19x + 60 = (x - 15)(x - 4)
So,
P(x) = -2(x - 15)(x - 4)
Now we need to find the range of x where P(x) > 0. To do this, we can look at the sign of each factor in the expression:
1. -2 is always negative.
2. (x - 15) is positive if x > 15, and negative if x < 15.
3. (x - 4) is positive if x > 4, and negative if x < 4.
We have three cases to consider:
Case 1: x < 4
In this case, both (x - 15) and (x - 4) are negative. A negative times a negative is a positive, so the product (x - 15)(x - 4) is positive. But since we also have a factor of -2, which is negative, the entire expression P(x) = -2(x-15)(x-4) would be negative. This means that the company will not make a profit for x < 4.
Case 2: 4 < x < 15
In this case, (x - 15) is negative and (x - 4) is positive. A negative times a positive is a negative, so the product (x - 15)(x - 4) is negative. But since we also have a factor of -2, which is negative, the entire expression P(x) = -2(x-15)(x-4) would be positive. This means that the company will make a profit for 4 < x < 15.
Case 3: x > 15
In this case, both (x - 15) and (x - 4) are positive. A positive times a positive is a positive, so the product (x - 15)(x - 4) is positive. But since we also have a factor of -2, which is negative, the entire expression P(x) = -2(x-15)(x-4) would be negative. This means that the company will not make a profit for x > 15.
In summary, the company makes a profit for x in the range (4, 15), meaning when they produce more than 4,000 units but less than 15,000 units.