Your class is selling boxes of flower seeds as a fundraiser. The total profit p depends on the amount x that your class charges for each box of seeds. The equation p =-0.5x^2+36x-225 models the profit of the fundraiser. What's the smallest amount, in dollars, that you can charge and make a profit of at least $369?
To find the smallest amount that can be charged and make a profit of at least $369, we need to set up the inequality:
p ≥ 369
Substituting the equation for p, we have:
-0.5x^2 + 36x - 225 ≥ 369
Rearranging the inequality to standard quadratic form, we get:
-0.5x^2 + 36x - 594 ≥ 0
To solve this inequality, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.5, b = 36, and c = -594:
x = (-(36) ± √((36)^2 - 4(-0.5)(-594))) / (2(-0.5))
Simplifying this equation, we have:
x = (-36 ± √(1296 - 1188)) / (-1)
x = (-36 ± √(108)) / (-1)
x = (-36 ± 6√3) / (-1)
The discriminant (√(108) = 6√3 *2) is positive, so there are two distinct real solutions:
x = (-36 + 6√3) / (-1) and x = (-36 - 6√3) / (-1)
To find the smallest amount of dollars, x, we need to evaluate these two expressions.
1. x = (-36 + 6√3) / (-1):
≈ 42.94
2. x = (-36 - 6√3) / (-1):
≈ -5.94
The smallest amount that can be charged to make a profit of at least $369 is approximately $42.94.