The total profit function, P(x), for 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. [Hint: The company makes a profit when P(x)>0.] Explain and justify your answer.
To determine the values of x for which the company makes a profit, we need to find the values of x that satisfy the inequality P(x) > 0, where P(x) is the profit function.
The given profit function is P(x) = -2x^2 + 38x - 120.
To solve the inequality P(x) > 0, we can start by finding the x-intercepts of the quadratic equation -2x^2 + 38x - 120 = 0.
Setting P(x) = 0, we have -2x^2 + 38x - 120 = 0.
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
However, for the purpose of finding the values of x for which the company makes a profit (P(x) > 0), we only need to determine the x-intercepts.
Let's factor the quadratic equation:
-2x^2 + 38x - 120 = 0
-2(x^2 - 19x + 60) = 0
-2(x - 15)(x - 4) = 0
This equation has two x-intercepts: x = 15 and x = 4.
Now, let's analyze the profit function P(x) = -2x^2 + 38x - 120.
Since the coefficient of x^2 term is negative (-2), the parabola opens downwards. This means that the profit function is a quadratic function with a maximum value.
Thus, the company makes a profit when the function is positive, P(x) > 0, which corresponds to the values of x between the two x-intercepts (4 and 15).
Therefore, the values of x for which the company makes a profit are x > 4 and x < 15.
To find the values of x for which the company makes a profit, we need to determine when the profit function P(x) is greater than zero (P(x) > 0).
Given the profit function: P(x) = -2x^2 + 38x - 120.
To solve P(x) > 0, we can solve the quadratic inequality -2x^2 + 38x - 120 > 0.
To do that, we can use different methods, but one common method is to factorize the quadratic equation if possible.
However, in this case, the quadratic equation cannot be easily factorized. So, we'll use an alternate method known as the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are coefficients of the quadratic equation ax^2 + bx + c = 0.
In our case, the coefficients are as follows:
a = -2 (coefficient of x^2),
b = 38 (coefficient of x),
c = -120 (constant term).
Using the quadratic formula, we can find the values of x that satisfy the inequality -2x^2 + 38x - 120 > 0.
x = (-38 ± √(38^2 - 4 * -2 * -120)) / (2 * -2).
Simplifying further, we have:
x = (-38 ± √(1444 - 960)) / (-4).
x = (-38 ± √484) / (-4).
x = (-38 ± 22) / (-4).
So, we have two solutions:
x₁ = (-38 + 22) / (-4) = -16 / (-4) = 4.
x₂ = (-38 - 22) / (-4) = -60 / (-4) = 15.
Therefore, the company makes a profit when the variable x is between 4 and 15 (inclusive), which means x = 4, 5, 6, ..., 15.
To justify this answer, we can substitute these values back into the profit function P(x). If the result is greater than zero, it means the company makes a profit.
−2x^2+38x−120
P=0 at x=4,15
Since the parabola opens downward, the vertex is above the x-axis., P(x) is positive between these values.