Math Problem Statement

The profit P (in dollars) from selling x units of a product is given by the function below. P = 36,000 + 2051

x − 1 8x2 , 150 ≤ x ≤ 275 Find the marginal profit for each of the following sales. (Round your answers to two decimal places.)

Solution

To calculate the marginal profit, we need to find the derivative of the profit function P(x)P(x), which gives us the rate of change of profit with respect to the number of units sold xx.

Given the profit function:

P(x)=36,000+2051x18x2P(x) = 36,000 + 2051x - \frac{1}{8}x^2

The marginal profit is represented by the derivative P(x)P'(x). Let's differentiate P(x)P(x) with respect to xx:

P(x)=ddx(36,000+2051x18x2)P'(x) = \frac{d}{dx}(36,000 + 2051x - \frac{1}{8}x^2)

Step-by-step differentiation:

  1. The derivative of the constant 36,00036,000 is 0.
  2. The derivative of 2051x2051x is 20512051.
  3. The derivative of 18x2-\frac{1}{8}x^2 is 14x-\frac{1}{4}x (using the power rule).

Thus, the marginal profit function is:

P(x)=205114xP'(x) = 2051 - \frac{1}{4}x

Now, we can calculate the marginal profit at specific values of xx by plugging them into P(x)P'(x). Let me know which specific values you would like to calculate the marginal profit for, or if you have any predefined values.

Would you like to see detailed examples for specific xx values? Here are some related questions for practice:

  1. What is the marginal profit when x=200x = 200?
  2. How does the marginal profit change as xx increases?
  3. What is the interpretation of negative marginal profit?
  4. At which point is the marginal profit zero?
  5. How does this quadratic form affect the marginal profit compared to a linear profit model?

Tip: The marginal profit function can help determine the optimal production level by identifying where profit maximization occurs.

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Math Problem Analysis

Mathematical Concepts

Calculus
Marginal Profit
Derivatives
Quadratic Functions

Formulas

P(x) = 36,000 + 2051x - (1/8)x^2
P'(x) = 2051 - (1/4)x

Theorems

Power Rule of Differentiation

Suitable Grade Level

Grades 11-12 or College Level