Trouble solving equation, finding dimensions of rectangles with shared side (given perimeter)

Question

You will build a rectangular sheep pen next to a river. There is no need to build a fence along the river, so you only need to build three sides. You have a total of 450 feet of fence to use. Find the dimensions of the pen such that you can enclose the maximum area.

Answer 1

The way to solve this is to set $x$ to be the width of the rectangle and $y$ to be the length.

Therefore: $2x+y=450$, because of the restriction on the perimeter. $xy=$ the area.

Let us solve for $y$ in terms of $x$.

$y=450-2x$

Therefore, our area is $x(450-2x)$ or $-2x^2+450x$. Since this is an upside-down parabola (the negative coefficient on the $x^2$-value), we can use $-\frac{b}{2a}$ to find the maximum.

Our $b$ = 450, and $a=-2$, so we have the maximum area at the $x$-value:

$$x=-\frac{450}{-4}$$ $$x=112.5$$ By our first definition: $y=450-2x$

$$y=450-2 \cdot 112.5$$ $$y=225$$

Our maximum dimensions are length $225$ and width $112.5$, with area being $25312.5$.

Answer 2

We differentiate function of area wrt a and equating it with 0 SO that we can calc. a for which area will be max.