Question

suppose that you have 280 feet of rope and want to use it to make a rectangle. what dimensions should you make your rectangle if you want to enclose the maximum possible area? the length should be feet the width should be feet the total area enclosed is square feet.

Answer 1

To maximize the area of a rectangle with a fixed perimeter of 280 feet, one should construct a square with each side measuring 70 feet, giving an enclosed area of 4900 square feet.

Step-by-step explanation:

Maximizing the Area of a Rectangle with a Given Perimeter

To enclose the maximum possible area using a given amount of perimeter, which in this case is 280 feet of rope, the rectangle should be as close to a square as possible because a square has the largest area for a given perimeter among all rectangles. We can show this through calculus or by recognizing that the square is a special case of a rectangle where all sides are equal. In our case, the perimeter of the rectangle must be 280 feet, so if we let the sides of the square be s feet, then the perimeter is 4s = 280 feet. Solving for s we get 280/4 = 70 feet. Therefore, for maximum area, the rectangle should have dimensions of 70 feet by 70 feet which encloses an area of 4900 square feet.

If you cannot construct a perfect square, then the dimensions should be made as close to each other as possible. However, since we are working with a fixed perimeter and the question seeks the maximum area, the dimensions that would yield the largest area would indeed be a square with equal sides.