Question

The profit for a product can be described by the function ​p(x)=212x−5500−x2 dollars, where x is the number of units produced and sold. to maximize​ profit, how many units must be produced and sold?

a. 212 units
b. 100 units
c. 50 units
d. 106 units

Answer 1

To maximize profit for the given function p(x) = 212x - 5500 - x2, we calculate the vertex of the parabola, resulting in 106 units as the number of units to produce and sell to achieve maximum profit.

Step-by-step explanation:

To determine the number of units that must be produced and sold to maximize profit, we need to find the maximum point of the profit function p(x) = 212x - 5500 - x2 dollars, where x is the number of units. We can find this maximum by calculating the vertex of the parabola represented by the profit function because the profit function is a quadratic equation and its graph will be a parabola opening downwards (since the coefficient of x2 is negative).

The vertex form of a parabola is given by p(x) = a(x-h)2 + k, where (h, k) is the vertex of the parabola. For a parabola ax2 + bx + c, the x-coordinate of the vertex (h-value) is determined by the formula -b / (2a). In our function p(x), a = -1 and b = 212, so we calculate h = -b / (2a) = -212 / (2 * -1) = 106. Therefore, the number of units that must be produced and sold to maximize profit is 106 units, corresponding to option d.