find the dimensions of a rectangle with a perimeter of 180 feet that has the maximum area. the side lengths…

Explanation:

Step1: Set up the equations

Let the length of the rectangle be (x) and the width be (y). The perimeter formula is (P = 2(x + y)), and given (P=180), so (2(x + y)=180), which simplifies to (x + y=90), and (y = 90 - x). The area formula is (A=xy). Substitute (y = 90 - x) into the area formula: (A(x)=x(90 - x)=90x - x^{2}).

Step2: Find the maximum of the area function

Since (A(x)=-x^{2}+90x) is a quadratic function of the form (y = ax^{2}+bx + c) ((a=-1), (b = 90), (c = 0)). For a quadratic function (y = ax^{2}+bx + c), the vertex (x) - coordinate is given by (x=-\frac{b}{2a}). Substitute (a=-1) and (b = 90) into (x=-\frac{b}{2a}), we get (x=-\frac{90}{2\times(-1)} = 45). Since (y=90 - x), when (x = 45), (y=45).

Answer:

(45,45)