Answer:
[tex]\text{Dimensions: 25 x 50},\\\text{Area: }1,250\:\mathrm{m^2}[/tex]
Step-by-step explanation:
Let the one of the side lengths of the rectangle be [tex]x[/tex] and the other be [tex]y[/tex].
We can write the following equations, where [tex]x[/tex] will be the side opposite to the wall:
[tex]x+2y=100,\\xy=\text{Area}[/tex]
From the first equation, we can isolate [tex]x=100-2y[/tex] and substitute into the second equation:
[tex](100-2y)y=\text{Area},\\-2y^2+100=\text{Area}[/tex]
Therefore, the parabola [tex]-2y^2+100y[/tex] denotes the area of this rectangular enclosure. The maximum area possible will occur at the vertex of this parabola.
The x-coordinate of the vertex of a parabola in standard form [tex]ax^2+bx+c[/tex] is given by [tex]\frac{-b}{2a}[/tex].
Therefore, the vertex is:
[tex]\frac{-100}{2(-2)}=\frac{100}{4}=25[/tex]
Plug in [tex]x=25[/tex] to the equation to get the y-coordinate:
[tex]-2(25^2)+100(25)=\boxed{1,250}[/tex]
Thus the vertex of the parabola is at [tex](25, 1250)[/tex]. This tells us the following:
- The maximum area occurs when one side (y) of the rectangle is equal to 25
- The maximum area of the enclosure is 1,250 square meters
- The other dimension, from [tex]x+2y=100[/tex], must be [tex]50[/tex]
And therefore, the desired answers are:
[tex]\text{Dimensions: 25 x 50},\\\text{Area: }1,250\:\mathrm{m^2}[/tex]
