find the dimensions of a rectangle with a perimeter of 200 feet that has the maximum area. the side lengths…

Explanation:

Step1: Define 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=200), so (2(x + y)=200), which simplifies to (x + y=100), and (y = 100 - x).

Step2: Express the area function

The area formula of a rectangle is (A=xy). Substitute (y = 100 - x) into the area formula, we get (A(x)=x(100 - x)=100x - x^{2}).

Step3: Find the derivative of the area function

Using the power rule ((x^{n})^\prime=nx^{n - 1}), the derivative (A^\prime(x)=(100x - x^{2})^\prime=100-2x).

Step4: Find the critical points

Set (A^\prime(x) = 0), so (100-2x = 0). Solving for (x): [ \begin{align*} 100-2x&=0\ 2x&=100\ x& = 50 \end{align*} ]

Step5: Check the second - derivative (to confirm it's a maximum)

The second - derivative (A^{\prime\prime}(x)=(100 - 2x)^\prime=-2\lt0). Since (A^{\prime\prime}(x)\lt0) when (x = 50), the function (A(x)) has a maximum at (x = 50). When (x = 50), then (y=100 - x=50)

Answer:

(50,50)