A rectangle has a perimeter of 40 cm. What dimensions will maximize the area? (2

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

A rectangle has a perimeter of 40 cm. What dimensions will maximize the area? (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 cm, 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 dimensions. The area A of a rectangle is calculated as A = length × width.
Step 5: To maximize the area, we can use the fact that a square has the maximum area for a given perimeter. This means length and width should be equal.
Step 6: Since length + width = 20 and both are equal, we can say length = width = x. So, x + x = 20.
Step 7: Solve for x: 2x = 20, which gives x = 10 cm.
Step 8: Therefore, the dimensions that maximize the area are length = 10 cm and width = 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 largest area.