Laura owns and operates Aunt Linda's Pecan Pies. She has learned that her profits, P(x), from the sale of x cases of pies, are given by P(x) = -x^2 + 14x + 51.
a. The company will "break-even" when the profit is zero. How many cases of pies should Laura sell in order to break-even? (Solve for x when P(x) = 0.)
x=17 or x=-3
To determine the number of cases Laura should sell in order to break-even, we need to solve the equation P(x) = 0, where P(x) represents the profit from selling x cases.
Given that P(x) = -x^2 + 14x + 51, we can substitute 0 for P(x) and solve for x:
0 = -x^2 + 14x + 51
Rearranging the equation, we get:
x^2 - 14x - 51 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. Factoring is not easily possible in this case, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -14, and c = -51. Substituting these values into the formula, we get:
x = (-(-14) ± √((-14)^2 - 4(1)(-51))) / (2(1))
x = (14 ± √(196 + 204)) / 2
x = (14 ± √400) / 2
x = (14 ± 20) / 2
Now we have two possible solutions:
x1 = (14 + 20) / 2 = 34 / 2 = 17
x2 = (14 - 20) / 2 = -6 / 2 = -3
Since the number of cases cannot be negative in this context, the only valid solution is x = 17.
Therefore, Laura should sell 17 cases of pies in order to break-even.
To find the number of cases Laura needs to sell in order to break-even, we need to solve the equation P(x) = 0, where P(x) represents the profit.
Given that P(x) = -x^2 + 14x + 51, we can substitute 0 for P(x) in the equation and solve for x:
0 = -x^2 + 14x + 51
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring may not be straightforward, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -1, b = 14, and c = 51. Substituting these values into the formula, we get:
x = (-14 ± √(14^2 - 4(-1)(51))) / (2(-1))
Simplifying further:
x = (-14 ± √(196 + 204)) / -2
x = (-14 ± √400) / -2
x = (-14 ± 20) / -2
We have two possible solutions:
1. x = (-14 + 20) / -2 = 6 / -2 = -3
2. x = (-14 - 20) / -2 = -34 / -2 = 17
Since selling negative cases of pies doesn't make sense in this context, we discard the solution x = -3.
Therefore, Laura needs to sell 17 cases of pies in order to break-even.