Math Problem Statement

A company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the given equation. Using this equation, find out what price the widgets should be sold for, to the nearest cent, for the company to make the maximum profit. y=−10x^2+692x−6342

Solution

To determine the price, xx, that maximizes the company's profit, we need to find the vertex of the quadratic function given by:

y=10x2+692x6342y = -10x^2 + 692x - 6342

For a quadratic equation in the form y=ax2+bx+cy = ax^2 + bx + c, the vertex occurs at:

x=b2ax = \frac{-b}{2a}

Here, a=10a = -10, b=692b = 692, and c=6342c = -6342.

Substitute these values into the formula for the vertex:

x=6922(10)=69220=34.6x = \frac{-692}{2(-10)} = \frac{-692}{-20} = 34.6

Thus, the price that maximizes profit is $34.60.

Answer: The company should sell each widget for $34.60 to make the maximum profit.

Would you like a detailed explanation of the process? Feel free to ask any follow-up questions!

Additional questions:

  1. How does the quadratic equation relate to profit maximization?
  2. What happens if the price is set too high or too low?
  3. How can we calculate the maximum profit using this price?
  4. What are other ways to represent quadratic equations for real-world situations?
  5. How would the equation change if the company wants to increase profits by a fixed amount?

Tip: The vertex formula is a quick way to find the point of maximum or minimum value in any quadratic function, which is useful in many optimization problems.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Profit Maximization
Vertex of a Quadratic Function

Formulas

Quadratic equation formula: y = ax^2 + bx + c
Vertex formula: x = -b / 2a

Theorems

Vertex Theorem for Quadratic Functions

Suitable Grade Level

Grades 9-11