Question

Suppose that you have 520 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?

a) The length should be 260 feet, The width should be 130 feet, The total area enclosed is 33,800 square feet.

b) The length should be 130 feet, The width should be 260 feet, The total area enclosed is 33,800 square feet.

c) The length should be 173.33 feet, The width should be 173.33 feet, The total area enclosed is 29,961.11 square feet.

d) The length should be 130 feet, The width should be 130 feet, The total area enclosed is 16,900 square feet.

Answer 1

To enclose the maximum possible area with the given 520 feet of rope, the rectangle should have dimensions of 130 feet by 130 feet, resulting in a total area of 16,900 square feet.

Step-by-step explanation:

To find the dimensions that will enclose the maximum possible area with the given 520 feet of rope, we can use the formula for the perimeter of a rectangle: P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. With 520 feet of rope, we can set up the equation 2L + 2W = 520. To find the maximum area, we can solve for one variable in terms of the other and substitute it into the area formula: A = LW. Solving the equation 2L + 2W = 520 for W, we get W = 260 - L. Substituting this into the area formula, we get A = L(260 - L), or A = 260L - L^2. To find the maximum area, we can complete the square or use calculus. Taking the derivative of A with respect to L and setting it equal to 0, we find that the maximum area occurs when L = 130 and W = 260 - L = 130. Therefore, the dimensions of the rectangle should be L = 130 feet and W = 130 feet, and the total area enclosed is 130 * 130 = 16,900 square feet. Therefore, the correct answer is d) The length should be 130 feet, The width should be 130 feet, and the total area enclosed is 16,900 square feet.