Question

The profit resulting from manufacturing and selling a product is represented by the function P(x) = -30(x - 500)² + 1000.

where x is the number of products manufactured and P(x) is the profit generated. How many items should be produced for
maximum profit?
O 200
O 1000
none of the answer choices
O 500
O There is no maximum profit.

Answer 1

Answer: The closest option among the given choices is:

O 500

Explanation:

To find the number of items that should be produced for maximum profit, we can analyze the given profit function P(x) = -30(x - 500)² + 1000.

The profit function is a quadratic function in the form of P(x) = ax² + bx + c, where:

a = -30

b = 30 * 500 * 2 = -30,000

c = 1000

Since the coefficient of the x² term (a) is negative, the quadratic function represents a downward-opening parabola. The vertex of this parabola represents the maximum point.

The x-coordinate of the vertex can be found using the formula: x = -b / (2a).

Substituting the values, we have:

x = -(-30,000) / (2 * -30)

x = 500

Therefore, the maximum profit will be achieved when 500 items are produced.