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 the profit is greater than or equal to $2300, we need to solve the equation -0.8x^2 + 106x - 1000 ≥ 2300.
First, we subtract 2300 from both sides of the inequality to get:
-0.8x^2 + 106x - 1000 - 2300 ≥ 0
Simplifying the equation, we have:
-0.8x^2 + 106x - 3300 ≥ 0
To solve this quadratic inequality, we can find its roots by setting it equal to zero:
-0.8x^2 + 106x - 3300 = 0
To solve this equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.8, b = 106, and c = -3300:
x = (-106 ± √(106^2 - 4(-0.8)(-3300))) / (2(-0.8))
x = (-106 ± √(11236 - 10560)) / (-1.6)
x = (-106 ± √676) / (-1.6)
x = (-106 ± 26) / (-1.6)
Now we have two possible solutions for x:
x1 = (-106 + 26) / (-1.6) = 80 / (-1.6) = -50
x2 = (-106 - 26) / (-1.6) = -132 / (-1.6) = 82.5
Since selling prices cannot be negative, the valid selling price for which the profit is greater than or equal to $2300 is x = 82.5 dollars.