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= -16x^2 + 124x + 110

To find the maximum profit, we need to find the vertex of the quadratic equation:

\[
y = -16x^2 + 124x + 110
\]

This equation represents a parabola that opens downwards (since the coefficient of $x^2$ is negative), meaning the vertex will give the maximum point.

### Step 1: Find the x-coordinate of the vertex
The formula for the x-coordinate of the vertex of a quadratic equation $ax^2 + bx + c$ is:

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

Here, $a = -16$ and $b = 124$:

\[
x = -\frac{124}{2(-16)} = \frac{124}{32} = 3.875
\]

### Step 2: Find the y-coordinate (maximum profit)
Now, substitute $x = 3.875$ back into the equation to find $y$:

\[
y = -16(3.875)^2 + 124(3.875) + 110
\]

Calculating step-by-step:

1. $(3.875)^2 = 15.015625$
2. $-16 \times 15.015625 = -240.25$
3. $124 \times 3.875 = 480.5$

Thus,

\[
y = -240.25 + 480.5 + 110 = 350.25
\]

The maximum profit is approximately $350.

### Answer:
The maximum profit the company can make is $350.

Do you want further details or have any questions?

Here are 5 related questions to explore:
1. How would the maximum profit change if the coefficient of $x^2$ were different?
2. What does the selling price $x = 3.875$ signify in a real-world context?
3. How can the vertex form of a quadratic equation be used to find the maximum or minimum values?
4. How does changing the constant term $c$ affect the maximum profit in a quadratic equation?
5. What are other methods to find the maximum value of a quadratic equation?

**Tip:** When dealing with quadratic equations, always check if the parabola opens upwards or downwards to determine whether you're finding a maximum or minimum value.

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 = -16x^2 + 124x + 110

Learn how to find the maximum profit a company can make by using the quadratic function y = -16x^2 + 124x + 110. This solution covers key algebraic concepts such as quadratic functions, vertex formulas, and optimization techniques. Suitable for students in Grades 9-12.

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