To find the maximum profit, we need to determine the vertex of the quadratic equation $y = -x^2 + 84x - 630$. The vertex of a parabola described by $y = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$.

Here:

• $a = -1$
• $b = 84$
• $c = -630$

Step 1: Find $x$ at the vertex

$x = -\frac{b}{2a} = -\frac{84}{2(-1)} = \frac{84}{2} = 42$

Step 2: Substitute $x = 42$ into the equation to find $y$

$y = -(42)^2 + 84(42) - 630$

Calculate each term: $-(42)^2 = -1764$ $84 \times 42 = 3528$ $y = -1764 + 3528 - 630$ $y = 1134$

Final Answer:

The maximum profit is $1,134.

Would you like a detailed breakdown of any step or have additional questions?

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5 Related Questions:

1. What is the significance of the vertex in real-world applications like profit maximization?
2. How does the coefficient of $x^2$ affect the shape of the graph of this profit function?
3. What does the $x$-coordinate of the vertex represent in this scenario?
4. Could you find the selling price of each widget that yields the maximum profit?
5. How would the equation change if fixed costs increased by $100?

Tip: For profit optimization problems, always check whether the quadratic function opens upwards ($a > 0$) or downwards ($a < 0$) to determine whether you're finding a maximum or minimum.