Question

riley wants to enclose an outdoor rectangular garden where one side of the enclosure is bounded by their house. if they have 102 feet of fencing to work with for the remaining three sides of the garden, find the dimensions of the rectangle that will result in a maximum area and find the maximum area. side length opposite the house: side length perpendicular to the house: maximum area:

Explanation:

Step1: Define variables

Let $x$ be the side - length perpendicular to the house and $y$ be the side - length opposite the house. We know that the amount of fencing is $2x + y=102$, so $y = 102 - 2x$.

Step2: Set up area function

The area $A$ of the rectangular garden is $A=xy$. Substitute $y = 102 - 2x$ into the area formula, we get $A(x)=x(102 - 2x)=102x-2x^{2}$.

Step3: Find the derivative

Differentiate $A(x)$ with respect to $x$. Using the power rule, $A^\prime(x)=\frac{d}{dx}(102x - 2x^{2})=102-4x$.

Step4: Find critical points

Set $A^\prime(x) = 0$ to find the critical points. So, $102-4x = 0$. Solving for $x$ gives $4x=102$, and $x = 25.5$.

Step5: Find the second - derivative

Differentiate $A^\prime(x)$ to get the second - derivative $A^{\prime\prime}(x)=\frac{d}{dx}(102 - 4x)=-4$. Since $A^{\prime\prime}(x)<0$, when $x = 25.5$, the area function $A(x)$ has a maximum.

Step6: Find $y$

Substitute $x = 25.5$ into $y = 102 - 2x$. Then $y=102-2\times25.5 = 51$.

Step7: Find the maximum area

Substitute $x = 25.5$ and $y = 51$ into the area formula $A = xy$. So, $A=25.5\times51 = 1300.5$.

Answer:

Side length opposite the house: 51 feet
Side length perpendicular to the house: 25.5 feet
Maximum area: 1300.5 square feet