To find the maximum area that could be obtained by varying $x$, we need to determine the vertex of the quadratic equation. The quadratic equation given is:

$A = -3x^2 + 82x + 23$

In general, a quadratic equation of the form $A = ax^2 + bx + c$ has its maximum or minimum value at the vertex, which occurs at:

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

For the given equation:

• $a = -3$
• $b = 82$

Substituting these values into the vertex formula:

$x = \frac{-82}{2(-3)} = \frac{-82}{-6} = \frac{82}{6}$

Now, let's calculate $x$ and then find the corresponding area $A$ at this value of $x$.The length $x$ that maximizes the area is approximately $13.67$. The maximum area that can be obtained by varying $x$ is approximately $583.33$ square units, rounded to two significant figures.

Would you like more details, or do you have any other questions?

Here are five related questions you might find interesting:

1. How do you find the vertex of a quadratic equation in other contexts?
2. What does the discriminant of this quadratic equation tell you about the roots?
3. How would you find the $x$-intercepts of this quadratic function?
4. What would the graph of this quadratic function look like, and where would the maximum point be located?
5. How would you approach this problem if the quadratic term had a positive coefficient?

Tip: In quadratic equations, the sign of the coefficient of $x^2$ tells you whether the parabola opens upwards or downwards, which helps determine if the vertex is a maximum or minimum.