A manufacture of a certain commodity has estimated that her profit in thousands of dollars is given by the expression
-6x^2+30x-10
where x (in thousands) is the number of units produced. What production range will enable the manufacturer to realize a profit of a least $14,000 on the commodity?
-6X^2 + 30X -10 = 14,
-6X^2 + 30X - 24 = 0,
Divide3 both sides by -6:
X^2 - 5X + 4 = 0,
(X-4)(X-1) = 0.
X-4 = 0,
X = 4.
X-1 = 0,
X = 1.
Production Range: 1000 to 4000 Units.
To find the production range that will enable the manufacturer to realize a profit of at least $14,000 on the commodity, we need to set up an inequality using the given expression for profit.
The expression for profit is:
Profit = -6x^2 + 30x - 10
We want to find the range of x values where the profit is at least $14,000. Let's set up the inequality:
-6x^2 + 30x - 10 ≥ 14,000
To solve this inequality, we first rearrange it to bring all terms to one side:
-6x^2 + 30x - 10 - 14,000 ≥ 0
Combine like terms:
-6x^2 + 30x - 14,010 ≥ 0
Now, we need to solve this quadratic inequality. There are different methods for solving quadratic inequalities, but one common approach is to find the x-values where the quadratic is equal to zero and then determine the intervals where it is positive or negative.
Let's factor the quadratic expression:
-6x^2 + 30x - 14,010 = (-6)(x - 83)(x + 35)
Set each factor equal to zero:
x - 83 = 0 or x + 35 = 0
Solve for x:
x = 83 or x = -35
Now, we have the critical points. We can choose test points in each interval and evaluate the inequality to determine the sign:
For x < -35, let's choose x = -40:
-6(-40)^2 + 30(-40) - 14,010 = -96,090
The result is negative, so this interval is not part of the solution.
For -35 < x < 83, let's choose x = 0:
-6(0)^2 + 30(0) - 14,010 = -14,010
The result is negative, so this interval is not part of the solution.
For x > 83, let's choose x = 90:
-6(90)^2 + 30(90) - 14,010 = 34,890
The result is positive, so this interval is part of the solution.
Therefore, the production range that will enable the manufacturer to realize a profit of at least $14,000 is x > 83 (in thousands of units produced).
To find the production range that will enable the manufacturer to realize a profit of at least $14,000, we need to set up an inequality using the given expression for profit.
The expression for profit is -6x^2 + 30x - 10, which represents profit in thousands of dollars.
We want the profit to be at least $14,000, so we can write the inequality as:
-6x^2 + 30x - 10 ≥ 14
To solve this inequality, let's first rearrange it:
-6x^2 + 30x - 10 - 14 ≥ 0
-6x^2 + 30x - 24 ≥ 0
Now let's factor out the common factor of -6:
-6(x^2 - 5x + 4) ≥ 0
Next, let's continue factoring the quadratic equation inside the parentheses:
-6(x - 1)(x - 4) ≥ 0
Now we have factored the inequality, and we have three critical points: x = 1, x = 4, and the point between these two values.
To determine the sign of the inequality in each interval, we can use a sign chart:
Interval 1: (-∞, 1)
Choose a test point, e.g., x = 0, and substitute it into the inequality:
-6(0 - 1)(0 - 4) ≥ 0
-6(1)(-4) ≥ 0
24 ≥ 0
Since 24 is positive, the inequality is true in this interval.
Interval 2: (1, 4)
Choose a test point, e.g., x = 2, and substitute it into the inequality:
-6(2 - 1)(2 - 4) ≥ 0
-6(1)(-2) ≥ 0
12 ≥ 0
Since 12 is positive, the inequality is true in this interval as well.
Interval 3: (4, ∞)
Choose a test point, e.g., x = 5, and substitute it into the inequality:
-6(5 - 1)(5 - 4) ≥ 0
-6(4)(1) ≥ 0
-24 ≥ 0
Since -24 is negative, the inequality is false in this interval.
Therefore, the solution to the inequality is the union of the intervals where the inequality is true, which is:
(-∞, 1] U (1, 4]
In the context of the problem, this means that the production range that will enable the manufacturer to realize a profit of at least $14,000 is between 0 and 1 thousand units produced, inclusive, and between 1 and 4 thousand units produced, exclusive.