The problem provided is a quadratic word problem, and the equation given is:

$y = -x^2 + 75x - 636$

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

• $a = -1$
• $b = 75$
• $c = -636$

Finding the Maximum Profit

Since the quadratic term $-x^2$ has a negative coefficient, the parabola opens downward, meaning it has a maximum point at the vertex. The formula to find the x-coordinate of the vertex for a quadratic equation $ax^2 + bx + c$ is:

$x = -\frac{b}{2a}$

Substituting the values of $a$ and $b$:

$x = -\frac{75}{2(-1)} = \frac{75}{2} = 37.5$

Now that we have $x = 37.5$, we can substitute this value back into the original equation to find the corresponding maximum profit $y$:

$y = -(37.5)^2 + 75(37.5) - 636$

Let's calculate this value to determine the maximum profit.The maximum profit the company can make is approximately $770, rounded to the nearest dollar.

Would you like more details or further clarification?

Here are 5 related questions to consider:

1. How do you find the vertex of a quadratic equation graphically?
2. What does it mean for a parabola to "open downwards" in terms of its maximum or minimum value?
3. How can you identify if a quadratic equation has a maximum or minimum value just by looking at the equation?
4. What is the significance of the coefficients $a$, $b$, and $c$ in a quadratic equation?
5. How does the discriminant of a quadratic equation relate to its solutions?

Tip: The x-value of the vertex in a quadratic equation represents the point where the maximum or minimum occurs, depending on the direction of the parabola.