Math Problem Statement

A local sneaker company’s profits π each unit of x sneaker sold has been estimated as π(x)= 〖-x〗^2+300x-9000 a)Find the number of sneaker sold in order for the company to maximize its profit. b)What is the maximum level of profit can the company achieve? c)Find the number of sneaker that the company have to sell in order to breakeven. d)Sketch the graph for the profit function for this company.

Solution

Here's the graph for the profit function π(x)=x2+300x9000\pi(x) = -x^2 + 300x - 9000:

  • The parabola opens downward.
  • The vertex, indicating the maximum profit, is at (150,13500)(150, 13500).
  • The breakeven points are approximately at x=34x = 34 and x=266x = 266.

Do you want details or have any questions?

Here are 5 questions you can ask next:

  1. How can I find the derivative of a function?
  2. What is the significance of the vertex in a quadratic function?
  3. Can you explain the quadratic formula in detail?
  4. How do you determine if a function is increasing or decreasing?
  5. What are some applications of quadratic functions in real life?

Tip: When working with quadratic functions, identifying the vertex and the direction in which the parabola opens can help you understand the function's maximum or minimum values and overall behavior.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex of a Parabola
Breakeven Point

Formulas

Quadratic function formula
Vertex formula for x-coordinate
Breakeven point formula

Theorems

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Suitable Grade Level

High School