Question

Lane Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can be approximated by the formula P = -4x2 + 24x + 3 where x is the number of units produced per week, in thousands.

a. How many units should the company produce per week to earn the maximum profit?
b. Find the maximum weekly profit.

Answer 1

Lane Manufacturing should produce 3,000 units per week for maximum profit, which is 39 hundred dollars or $3,900.

Step-by-step explanation:

To calculate the number of units Lane Manufacturing should produce for maximum profit, we need to find the vertex of the parabolic profit function given by P = -4x2 + 24x + 3. Since the coefficient of the x2 term is negative, the parabola opens downwards, causing the vertex to represent the maximum profit point.

The x-coordinate of the vertex of a parabola given by ax2 + bx + c is found using the formula -b / (2a). In our case, a = -4 and b = 24, so the x-coordinate is -24 / (2 * -4) = 3. Hence, Lane Manufacturing should produce 3,000 units per week to earn the maximum profit.

To find the maximum weekly profit, we substitute x = 3 into the profit function: P = -4(3)2 + 24(3) + 3 = -36 + 72 + 3 = 39 hundred dollars or $3,900.