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, equals, minus, 5, x, squared, plus, 209, x, minus, 1090
y=−5x 
2
 +209x−1090

The given equation for the company's profit, $ y $, as a function of the selling price of each widget, $ x $, is:

\[
y = -5x^2 + 209x - 1090
\]

This is a quadratic equation of the form $ y = ax^2 + bx + c $, where:

- $ a = -5 $
- $ b = 209 $
- $ c = -1090 $

### Finding the Price for Maximum Profit:
For a quadratic equation, the maximum (or minimum) value occurs at the **vertex**. Since $ a < 0 $ (i.e., $ a = -5 $), the parabola opens downwards, meaning there is a **maximum** profit at the vertex.

The formula for the $ x $-coordinate of the vertex is:

\[
x = \frac{-b}{2a}
\]

Substitute the values of $ a $ and $ b $ into this formula:

\[
x = \frac{-209}{2(-5)} = \frac{-209}{-10} = 20.9
\]

Thus, the price per widget that maximizes profit is **$20.90**.

### Conclusion:
The company should sell the widgets for **$20.90** to maximize its profit.

Would you like further details or any clarification?

---

Here are 5 related questions to deepen your understanding:
1. What is the maximum profit the company can make?
2. How would the profit change if the selling price is increased beyond $20.90?
3. What happens to the profit if the company decreases the price to $15?
4. How would the result change if the coefficient of $ x^2 $ were different?
5. Can the company still make a profit if the price is set at $10?

**Tip**: In any quadratic profit equation, when the coefficient of $ x^2 $ is negative, the maximum occurs at the vertex!

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 = -5x^2 + 209x - 1090

Discover how to determine the optimal selling price for widgets to maximize profit using quadratic equations. This problem involves understanding key algebraic concepts such as quadratic equations, profit maximization, and the vertex formula. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation