Question

A rectangular area is to be enclosed using an existing

Answer 1

Answer:

$\text{Dimensions: 25 x 50},\\\text{Area: }1,250\:\mathrm{m^2}$

Step-by-step explanation:

Let the one of the side lengths of the rectangle be $x$ and the other be $y$.

We can write the following equations, where $x$ will be the side opposite to the wall:

$x+2y=100,\\xy=\text{Area}$

From the first equation, we can isolate $x=100-2y$ and substitute into the second equation:

$(100-2y)y=\text{Area},\\-2y^2+100=\text{Area}$

Therefore, the parabola $-2y^2+100y$ 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 $ax^2+bx+c$ is given by $\frac{-b}{2a}$.

Therefore, the vertex is:

$\frac{-100}{2(-2)}=\frac{100}{4}=25$

Plug in $x=25$ to the equation to get the y-coordinate:

$-2(25^2)+100(25)=\boxed{1,250}$

Thus the vertex of the parabola is at $(25, 1250)$. 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 $x+2y=100$, must be $50$

And therefore, the desired answers are:

$\text{Dimensions: 25 x 50},\\\text{Area: }1,250\:\mathrm{m^2}$