A farmer wants to fence a rectangular field with 100 m of fencing. What dimensio

A farmer wants to fence a rectangular field with 100 m of fencing. What dimensions will maximize the area? (2022)

A farmer wants to fence a rectangular field with 100 m of fencing. What dimensions will maximize the area? (2022)

Step 1: Understand that the farmer has 100 meters of fencing to create a rectangular field.
Step 2: Remember that the perimeter of a rectangle is calculated as P = 2(length + width).
Step 3: Set the perimeter equal to 100 meters: 2(length + width) = 100.
Step 4: Simplify the equation: length + width = 50.
Step 5: To maximize the area of the rectangle, we need to find the best dimensions (length and width).
Step 6: The area of a rectangle is calculated as Area = length × width.
Step 7: Substitute width with (50 - length) in the area formula: Area = length × (50 - length).
Step 8: This gives us the area as a function of length: Area = 50length - length².
Step 9: Recognize that this is a quadratic equation that opens downwards, meaning it has a maximum point.
Step 10: The maximum area occurs when length = width, which means the rectangle is a square.
Step 11: Since length + width = 50, if both are equal, then length = width = 50/2 = 25 meters.
Step 12: Therefore, the dimensions that maximize the area are 25 meters by 25 meters.

Optimization of Area – The problem tests the understanding of how to maximize the area of a rectangle given a fixed perimeter.
Properties of Rectangles and Squares – It assesses knowledge of the relationship between the dimensions of rectangles and their areas, particularly that a square has the maximum area for a given perimeter.