Final answer:
The largest area of the rectangle with sides x and 18-x is found at the vertex of the parabola represented by the area function A = -x^2 + 18x. The vertex occurs at x = 9, resulting in a maximum area of 81 square units.
Step-by-step explanation:
To find the largest area of a rectangle with sides of x and 18-x, we can model the area A as a function of x by the equation A = x(18 - x). This gives us a quadratic function A = -x^2 + 18x, where the maximum value can be found by completing the square or using the vertex formula for a parabola. The vertex of the parabola will provide the maximum area since the leading coefficient is negative, indicating a downward opening parabola.
The vertex (h, k) of a parabola given by the function f(x) = ax^2 + bx + c is found at h = -b/(2a). Plugging the coefficients from our equation we get
h = -18 / (2 * -1) = 9
. We substitute
x = 9
back into the area function to find the maximum area,
A = 9 * (18 - 9) = 81 square units
. Therefore, the answer is
b) 81 square units
.