Maximum-area rectangles Of all rectangles with a fixed perimeter of P, which one has the maximum area? (Give the dimensions in terms of P.)

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Start by expressing the perimeter of the rectangle in terms of its length (l) and width (w). The perimeter P is given by the formula: P = 2l + 2w.

Solve the perimeter equation for one of the variables, say w, in terms of l and P: w = (P/2) - l.

Express the area A of the rectangle in terms of l and w. The area A is given by: A = l * w.

Substitute the expression for w from step 2 into the area formula: A = l * ((P/2) - l).

To find the maximum area, take the derivative of A with respect to l, set it to zero, and solve for l. This will give you the length that maximizes the area. Use the second derivative test to confirm that this value of l gives a maximum area.

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Perimeter and Area of Rectangles

The perimeter of a rectangle is the total distance around it, calculated as P = 2(l + w), where l is the length and w is the width. The area is given by A = l * w. Understanding the relationship between these two formulas is crucial for determining how to maximize the area while keeping the perimeter constant.

Optimization in Calculus

Optimization involves finding the maximum or minimum values of a function. In this context, we need to maximize the area function A(l) under the constraint of a fixed perimeter. This typically involves using techniques such as taking derivatives and applying critical point analysis to find where the area is maximized.

Critical Points and the First Derivative Test

Critical points occur where the first derivative of a function is zero or undefined. To find the maximum area of the rectangle, we differentiate the area function with respect to one variable, set the derivative to zero, and solve for that variable. The first derivative test helps determine whether these critical points correspond to a maximum or minimum.