Question

A company's profit can be modeled by the equation p(u) = -u^2 + 180u + 1000 where 'u' is the number of units sold. Find the maximum profit of the company. How many units should the company sell to maximize profit?

A. The maximum profit is $10,250, and the company should sell 90 units to maximize profit.
B. The maximum profit is $10,500, and the company should sell 100 units to maximize profit.
C. The maximum profit is $10,250, and the company should sell 100 units to maximize profit.
D. The maximum profit is $10,500, and the company should sell 90 units to maximize profit.

Answer 1

To find the maximum profit, we calculate the vertex of the profit parabola. The maximum number of units to sell is 90, which yields a profit of $10,100. Option A is closest but does not provide an accurate profit value.

Step-by-step explanation:

To maximize profit for the company using the profit function p(u) = -u2 + 180u + 1000, we first need to identify the vertex of the parabola represented by the function, since the highest point will give us the maximum profit and the number of units that the company should sell to achieve it. The general form of a quadratic function is f(x) = ax2 + bx + c. For a parabola that opens downwards (a negative 'a' value), the vertex formula for the x-coordinate is -b/(2a).

Here, a = -1 and b = 180, so the number of units to maximize profit is -b/(2a) = -180 / (2 * -1) = 90. Next, we calculate the maximum profit by substituting 'u' with 90 in the profit function: p(90) = -(90)2 + 180(90) + 1000 = -8100 + 16200 + 1000 = $10,100. Hence, the correct answer is none of the provided options, but if we had to choose the closest one, it would be Option A: The maximum profit is $10,250, and the company should sell 90 units to maximize profit. However, the profit value in Option A is not accurate based on our calculation.