Question

henry has 60 feet of fencing to surround his rectangular garden. what should the length and width be so that the garden will have the largest area? length = feet width = feet

Explanation:

Step1: Recall Perimeter and Area of Rectangle

For a rectangle, perimeter \( P = 2(l + w) \), area \( A = lw \). Given \( P = 60 \), so \( 2(l + w)=60 \), simplify to \( l + w = 30 \), so \( l = 30 - w \).

Step2: Express Area in Terms of Width

Substitute \( l = 30 - w \) into area formula: \( A = w(30 - w)=30w - w^2 \). This is a quadratic function \( A(w)= -w^2 + 30w \), which opens downward (since coefficient of \( w^2 \) is negative), so maximum at vertex.

Step3: Find Vertex of Quadratic

For quadratic \( ax^2 + bx + c \), vertex at \( w = -\frac{b}{2a} \). Here, \( a = -1 \), \( b = 30 \), so \( w = -\frac{30}{2(-1)} = 15 \).

Step4: Find Length

Since \( l = 30 - w \), substitute \( w = 15 \), get \( l = 30 - 15 = 15 \). So the rectangle is a square (which is a special rectangle) with length and width both 15.

Answer:

length = 15 feet, width = 15 feet