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

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

Options:

Correct Answer: 10, 10

Exam Year: 2022

Solution:

For a fixed perimeter, the area is maximized when the rectangle is a square. Thus, side = 10 units.

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

A rectangle has a perimeter of 40 units. 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 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 express the area (A) in terms of one variable. The area formula is A = length × width.
Step 5: From the equation length + width = 20, we can express width as width = 20 - length.
Step 6: Substitute this expression for width into the area formula: A = length × (20 - length).
Step 7: This gives us the area as a function of length: A = 20length - length².
Step 8: To find the maximum area, we can use calculus or recognize that this is a quadratic equation that opens downwards (since the coefficient of length² is negative).
Step 9: The maximum area occurs at the vertex of the parabola. The vertex for the equation A = -length² + 20length occurs at length = -b/(2a), where a = -1 and b = 20.
Step 10: Calculate the vertex: length = -20/(2 * -1) = 10 units.
Step 11: Since length + width = 20, if length = 10, then width = 20 - 10 = 10 units.
Step 12: Therefore, the dimensions that maximize the area are length = 10 units and width = 10 units, making it a square.

Perimeter and Area Relationship – Understanding how the perimeter of a rectangle relates to its area and how to optimize area for a fixed perimeter.
Properties of Rectangles and Squares – Recognizing that among all rectangles with a given perimeter, a square has the maximum area.