a college is planning to construct a rectangular parking lot on land bordered on one side by a highway. the…

Explanation:

Step1: Define variables

Let the width of the parking - lot be $x$ (the two equal sides perpendicular to the highway) and the length be $y$ (the side parallel to the highway). The total length of the fence is $2x + y=720$, so $y = 720 - 2x$.

Step2: Express the area function

The area $A$ of a rectangle is $A=xy$. Substitute $y = 720 - 2x$ into the area formula, we get $A(x)=x(720 - 2x)=720x-2x^{2}$.

Step3: Find the derivative of the area function

The derivative of $A(x)$ with respect to $x$ is $A^\prime(x)=\frac{d}{dx}(720x - 2x^{2})=720-4x$.

Step4: Find the critical points

Set $A^\prime(x)=0$, then $720 - 4x = 0$. Solving for $x$ gives $4x=720$, so $x = 180$.

Step5: Find the second - derivative of the area function

The second - derivative $A^{\prime\prime}(x)=\frac{d}{dx}(720 - 4x)=-4<0$. Since $A^{\prime\prime}(x)<0$ when $x = 180$, the function $A(x)$ has a maximum at $x = 180$.

Step6: Find the length

Substitute $x = 180$ into the equation $y = 720 - 2x$. Then $y=720-2\times180 = 360$.

Answer:

The width is 180 feet. The length is 360 feet.