Chapter 4 - Section 4.4 - Build Quadratic Models from Verbal Descriptions and From Data - 4.4 Assess Your Understanding - Page 308: 10

$1,000,000m^2$

The area enclosed is $x(2000-2x)=-2x^2+2000x$. Let's compare $f(x)=-2x^2+4000x$ to $f(x)=ax^2+bx+c$. We can see that a=-2, b=2000, c=0. $a\lt0$, hence the graph opens down, hence its vertex is a maximum. The maximum value is at $x=-\frac{b}{2a}=-\frac{2000}{2\cdot(-2)}=500.$ Hence the maximum value is $f(500)=-2(500)^2+2000(500)=1,000,000.$