刭 2 A real estate company owns a "defective rectangular" vacant land $A B C D E$, with side lengths and directions as shown in Figure 1. It is intended to build an apartment with a rectangular foundation running east-west on this land. Please draw the foundation of this land and calculate the maximum area of the foundation (accurate to $1 \mathrm{~m}^{2}$ ).

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A real estate company owns a "defective rectangular" vacant land $A B C D E$, with side lengths and directions as shown in Figure 1. It is intended to build an apartment with a rectangular foundation running east-west on this land. Please draw the foundation of this land and calculate the maximum area of the foundation (accurate to $1 \mathrm{~m}^{2}$ ).
Solution: Establish a rectangular coordinate system with line $BC$ and $AE$ as the $x$-axis and $y$-axis, respectively. The positive directions are along $BC$ and $AE$, with the unit of length being meters $(\mathrm{m})$.
The equation of line $AB$ is
$$
\frac{x}{30}+\frac{y}{20}=1 \text {. }
$$

Obviously, one vertex of the largest rectangle should be $D$. First, consider the case where the non-adjacent vertex $F$ is on $AB$. Then $F\left(x, 20-\frac{2}{3} x\right), 0 \leqslant x \leqslant 30$.
Thus, the area of the rectangle is
$$
\begin{aligned}
S & =(100-x)\left[80-\left(20-\frac{2}{3} x\right)\right] \\
& =-\frac{2}{3} x^{2}+\frac{20}{3} x+6000 \\
& =-\frac{2}{3}(x-5)^{2}+6016 \frac{2}{3} .
\end{aligned}
$$
$\therefore$ When $x=5, y=20-\frac{2}{3} x \approx 17$, the maximum
rectangular area is $6000 \mathrm{~m}^{2}$ and $5600 \mathrm{~m}^{2}$.
Answer: Select point $F(5,17)$, the maximum area is 6017 $\mathrm{m}^{2}$.