Question

find the dimensions of a rectangle with a perimeter of 124 feet that has the maximum area. the side lengths are (square) feet. (use a comma to separate answers as needed.)

Explanation:

Step1: Set up variables

Let the length of the rectangle be \(x\) and the width be \(y\). The perimeter formula is \(P = 2(x + y)\), and given \(P=124\), so \(2(x + y)=124\), which simplifies to \(x + y=62\), and \(y = 62 - x\). The area formula is \(A=xy=x(62 - x)=62x - x^{2}\).

Step2: Find the maximum of the area function

The area function \(A(x)=-x^{2}+62x\) is a quadratic function. For a quadratic function \(y = ax^{2}+bx + c\) (\(a=- 1\), \(b = 62\)), the vertex of the parabola (which gives the maximum for \(a<0\)) has \(x\) - coordinate \(x=-\frac{b}{2a}\).
Substitute \(a=-1\) and \(b = 62\) into \(x=-\frac{b}{2a}\), we get \(x=-\frac{62}{2\times(-1)} = 31\).
Since \(y=62 - x\), when \(x = 31\), \(y=62-31 = 31\).

Answer:

\(31,31\)