Question

find the length and width (in meters) of a rectangle that has the given perimeter and a maximum area. (enter your answers as a comma - separated list.) perimeter: 68 meters

Explanation:

Step1: Let variables

Let the length be \(x\) meters and the width be \(y\) meters. The perimeter \(P = 2(x + y)=68\), so \(x + y=34\), and \(y = 34 - x\).

Step2: Set up area function

The area \(A=xy=x(34 - x)=34x - x^{2}\).

Step3: Find derivative of area function

Differentiate \(A(x)\) with respect to \(x\). \(A^\prime(x)=\frac{d}{dx}(34x - x^{2})=34 - 2x\).

Step4: Find critical points

Set \(A^\prime(x) = 0\), \(34-2x = 0\), then \(2x=34\), \(x = 17\).

Step5: Second - derivative test

Differentiate \(A^\prime(x)\) to get \(A^{\prime\prime}(x)=- 2<0\). So when \(x = 17\), the area is maximized.

Step6: Find width

Since \(y=34 - x\), when \(x = 17\), \(y = 17\).

Answer:

\(17,17\)