The owner of a business that sells small handmade leather purses analyzes her production costs and studies the consumer demand for her purses. She creates an equation to model the results of her research: p = −0.8x^2 + 106x – 1000.
In this equation, p represents her profit in dollars and x represents the selling price in dollars. For what selling prices is her profit greater than or equal to $2300?
To find the selling prices for which her profit is greater than or equal to $2300, we need to find the values of x that satisfy the inequality p ≥ 2300.
Substituting p = −0.8x^2 + 106x – 1000 into the inequality, we get:
−0.8x^2 + 106x – 1000 ≥ 2300
Rearranging the inequality, we get:
−0.8x^2 + 106x – 4300 ≥ 0
To solve this quadratic inequality, we can factor the quadratic expression:
−0.8x^2 + 106x – 4300 = 0
Dividing the equation by -0.8 to make the leading coefficient positive, we get:
x^2 - 132.5x + 5375 = 0
Factoring the quadratic expression gives us:
(x - 25)(x - 215) ≥ 0
To find the values of x that satisfy this inequality, we consider the sign of the expression (x - 25)(x - 215) for different intervals:
Interval 1: x < 25
The expression (x - 25)(x - 215) is positive in this interval.
Interval 2: 25 ≤ x ≤ 215
The expression (x - 25)(x - 215) is negative in this interval.
Interval 3: x > 215
The expression (x - 25)(x - 215) is positive in this interval.
We are interested in the values of x for which the inequality is true (i.e., the expression (x - 25)(x - 215) is non-negative). Therefore, the solution to the inequality is:
x < 25 or x > 215
So, for selling prices less than 25 dollars or greater than 215 dollars, her profit will be greater than or equal to $2300.