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.
-6x^2+319x-2460

The profit function for the company is given by:

\[
y = -6x^2 + 319x - 2460
\]

This represents a quadratic equation of the form $ y = ax^2 + bx + c $, where $ a = -6 $, $ b = 319 $, and $ c = -2460 $.

To find the price $ x $ that maximizes the profit, we need to find the vertex of the parabola. For a quadratic equation in the form $ y = ax^2 + bx + c $, the $ x $-coordinate of the vertex, which gives the maximum profit, is given by:

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

### Step 1: Calculate the price that maximizes the profit
Substituting the values of $ a = -6 $ and $ b = 319 $ into the formula:

\[
x = -\frac{319}{2(-6)} = \frac{319}{12}
\]

Now let's calculate this.

\[
x \approx 26.58
\]

Thus, the price that maximizes the company's profit is approximately **$26.58**.

Would you like a detailed explanation of how this price affects the profit, or have any questions? 

Here are 5 related questions to deepen your understanding:
1. How does the sign of $ a $ (negative in this case) affect the shape of the parabola?
2. Can you calculate the maximum profit the company will make at $ x = 26.58 $?
3. How would the equation change if the company wanted to minimize a loss instead of maximizing profit?
4. What happens to the profit if the price is slightly above or below $26.58?
5. How can you apply this method to real-life business scenarios involving other costs?

**Tip:** In quadratic equations, if $ a $ is negative, the function opens downwards, indicating a maximum 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. -6x^2 + 319x - 2460

Discover how to maximize profit for a company selling widgets using the quadratic equation -6x^2 + 319x - 2460. Learn key algebraic concepts, including finding the vertex of a parabola to optimize outcomes. This guide provides a step-by-step solution suitable for high school students in Grades 10-12.

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