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.