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 the maximum amount of profit the company can make, to the nearest dollar.
y, equals, minus, 7, x, squared, plus, 584, x, minus, 5454
y=−7x 
2
 +584x−5454

To find the maximum profit, we need to determine the vertex of the quadratic equation. The equation given is:

\[
y = -7x^2 + 584x - 5454
\]

In a quadratic equation of the form $ y = ax^2 + bx + c $, the x-coordinate of the vertex, which gives the maximum value for downward-opening parabolas (when $ a < 0 $), is given by the formula:

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

Here, $ a = -7 $ and $ b = 584 $. Substituting these values into the formula:

\[
x = \frac{-584}{2(-7)} = \frac{-584}{-14} = 41.71
\]

Next, we substitute $ x = 41.71 $ back into the original equation to find the maximum profit:

\[
y = -7(41.71)^2 + 584(41.71) - 5454
\]

Let's calculate this step-by-step.

1. Calculate $(41.71)^2$.
2. Multiply the result by $-7$.
3. Calculate $584 \times 41.71$.
4. Add the results of steps 2 and 3, and then subtract 5454.

I'll compute this.The maximum profit the company can make is approximately **$6,727**, which occurs when the selling price of each widget is about $41.71.

Would you like more details on this calculation or have any other questions?

Here are some related questions you might consider:
1. What is the minimum profit the company can make?
2. How does changing the selling price affect the profit?
3. What are the roots of this profit equation?
4. How do you find the break-even points from the given profit equation?
5. How would a change in the cost of production affect the maximum profit?

**Tip:** The vertex formula $\frac{-b}{2a}$ is useful for finding maximum or minimum values of quadratic functions, depending on the sign of $ a $.

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: y = -7x^2 + 584x - 5454. Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar.

Learn how to find the maximum profit a company can make using the quadratic equation y = -7x^2 + 584x - 5454. This solution involves key algebraic concepts like the vertex formula and solving quadratic equations, suitable for Grades 10-12.

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