A rectangle has a perimeter of 40 cm. What dimensions maximize the area? (2022)

A rectangle has a perimeter of 40 cm. What dimensions maximize the area? (2022)

Step 1: Understand that the perimeter of a rectangle is the total distance around it. The formula for the perimeter (P) is P = 2(length + width).
Step 2: We know the perimeter is 40 cm, so we can write the equation: 2(length + width) = 40.
Step 3: Simplify the equation by dividing both sides by 2: length + width = 20.
Step 4: To maximize the area of the rectangle, we need to find the best dimensions. The area (A) of a rectangle is given by A = length × width.
Step 5: From the equation length + width = 20, we can express width in terms of length: width = 20 - length.
Step 6: Substitute this expression for width into the area formula: A = length × (20 - length).
Step 7: This gives us a quadratic equation for area: A = 20length - length².
Step 8: The maximum area of a rectangle occurs when it is a square. This means length = width.
Step 9: Since length + width = 20, if both are equal, we can say length = width = 20/2 = 10 cm.
Step 10: Therefore, the dimensions that maximize the area are 10 cm by 10 cm, making it a square.

Perimeter and Area of a Rectangle – Understanding the relationship between the perimeter of a rectangle and how to derive its dimensions to maximize the area.
Optimization in Geometry – Applying principles of optimization to determine the dimensions that yield the maximum area for a given perimeter.
Properties of Squares – Recognizing that among all rectangles with a given perimeter, a square has the maximum area.