Profit P(x) can be calculated as revenue minus cost. R(x) = 55x - 2x2 C(x) = 21x + 97 P(x) = - 2x2 + 34x - 97 Calculate the x-value that maximizes profit.
Mathematics · Middle School · Thu Feb 04 2021
Answered on
To find the x-value that maximizes the profit P(x), we need to find the vertex of the parabola represented by the profit function. Since the coefficient of \(x^2\) is negative (-2), the parabola opens downward, which means the vertex will give us the maximum point.
The profit function P(x) is given as: \[ P(x) = -2x^2 + 34x - 97 \]
The vertex of a parabola in the form of \( ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex, \( x_{vertex} = -\frac{b}{2a} \).
In this case, \( a = -2 \) and \( b = 34 \). Let's find the vertex: \[ x_{vertex} = -\frac{34}{2(-2)} = -\frac{34}{-4} = \frac{34}{4} = 8.5 \]
So, the x-value that maximizes the profit is \( x = 8.5 \).