A rectangle has a perimeter of 40 units. What dimensions maximize the area? (202

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

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

Step 1: Understand that the perimeter of a rectangle is the total distance around it, calculated as P = 2(length + width).
Step 2: We know the perimeter is 40 units, so we can set up 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 combination of length and width.
Step 5: The area of a rectangle is calculated as Area = length × width.
Step 6: To maximize the area, we can use the fact that a square (where length = width) has the largest area for a given perimeter.
Step 7: Since length + width = 20, if we set length = width, we can say length = width = 20/2 = 10.
Step 8: Therefore, the dimensions that maximize the area are length = 10 units and width = 10 units, 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 largest area.