a gardener has 1080 feet of fencing to fence in a rectangular garden. one side of the garden is bordered by…

Explanation:

Step1: Define variables

Let the side perpendicular to the river be $x$ (shorter side) and the side parallel to the river be $y$ (longer side). The perimeter equation is $2x + y=1080$, so $y = 1080 - 2x$.

Step2: Set up area function

The area of the rectangle $A=xy$. Substitute $y = 1080 - 2x$ into the area formula, we get $A(x)=x(1080 - 2x)=1080x-2x^{2}$.

Step3: Find the derivative

Differentiate $A(x)$ with respect to $x$. $A'(x)=\frac{d}{dx}(1080x - 2x^{2})=1080-4x$.

Step4: Find critical - points

Set $A'(x) = 0$. Then $1080-4x = 0$. Solving for $x$ gives $4x=1080$, so $x = 270$.

Step5: Find the second - derivative

Differentiate $A'(x)$ to get the second - derivative $A''(x)=\frac{d}{dx}(1080 - 4x)=-4<0$. Since $A''(x)<0$, when $x = 270$, the area is maximized.

Step6: Find the value of $y$

Substitute $x = 270$ into $y = 1080 - 2x$. Then $y=1080-2\times270 = 540$.

Step7: Calculate the maximum area

$A = xy=270\times540 = 145800$.

Answer:

shorter side: 270 ft longer side: 540 ft greatest possible area: 145800 $ft^{2}$