What is the maximum profit if the profit function is P(x) = -x^2 + 10x - 16? (20

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Question: What is the maximum profit if the profit function is P(x) = -x^2 + 10x - 16? (2021)

Options:

Correct Answer: 8

Exam Year: 2021

Solution:

The maximum profit occurs at x = -b/(2a) = 10/2 = 5. P(5) = -5^2 + 10*5 - 16 = 9.

What is the maximum profit if the profit function is P(x) = -x^2 + 10x - 16? (2021)

What is the maximum profit if the profit function is P(x) = -x^2 + 10x - 16? (2021)

Step 1: Identify the profit function, which is P(x) = -x^2 + 10x - 16.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = -1, b = 10, and c = -16.
Step 3: To find the maximum profit, use the formula for the x-coordinate of the vertex of a parabola, which is x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -10/(2 * -1).
Step 5: Calculate the value: x = 10/2 = 5.
Step 6: Now, substitute x = 5 back into the profit function to find the maximum profit: P(5) = -5^2 + 10*5 - 16.
Step 7: Calculate P(5): P(5) = -25 + 50 - 16.
Step 8: Simplify the calculation: P(5) = 50 - 25 - 16 = 9.
Step 9: Therefore, the maximum profit is 9.

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