Question

attempt 3: 8 attempts remaining. morgan wants to enclose an outdoor rectangular garden where one side of the enclosure is bounded by their house. if they have 110 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:
feet side length perpendicular to the house:
feet maximum area:
square feet video example: solving a similar problem

Explanation:

Step1: Define Variables

Let \( x \) be the side length perpendicular to the house, and \( y \) be the side length opposite the house. The total fencing used is \( 2x + y = 110 \), so \( y = 110 - 2x \).

Step2: Area Function

The area \( A \) of the rectangle is \( A = x \cdot y \). Substitute \( y \) from Step 1: \( A(x) = x(110 - 2x) = 110x - 2x^2 \).

Step3: Find Vertex of Parabola

The area function is a quadratic \( A(x) = -2x^2 + 110x \) with \( a = -2 \), \( b = 110 \). The \( x \)-coordinate of the vertex (maximum) is \( x = -\frac{b}{2a} = -\frac{110}{2(-2)} = \frac{110}{4} = 27.5 \).

Step4: Find \( y \)

Substitute \( x = 27.5 \) into \( y = 110 - 2x \): \( y = 110 - 2(27.5) = 110 - 55 = 55 \).

Step5: Maximum Area

\( A = x \cdot y = 27.5 \times 55 = 1512.5 \).

Answer:

• Side length opposite the house: \( \boldsymbol{55} \) feet
• Side length perpendicular to the house: \( \boldsymbol{27.5} \) feet
• Maximum area: \( \boldsymbol{1512.5} \) square feet