Final answer:
To find the dimensions of the rectangle with the maximum area, let's use the formulas for perimeter and area of a rectangle. First, we set up an equation using the information given and solve for one variable in terms of the other. Then, using the equation for the area of a rectangle, we find the quadratic equation that represents the area and determine the value of the variable that maximizes this area.
Step-by-step explanation:
To find the dimensions of the rectangle with the maximum area, we need to remember that the perimeter of a rectangle is twice the sum of its length and width. Let's say the length of the rectangle is 'L' and the width is 'W'. According to the problem, the perimeter is 124 feet, so we can write the equation as: 2(L + W) = 124. Solving for one variable in terms of the other, we get L = 62 - W. The formula for the area of a rectangle is A = L * W. Substituting the value of L, we get A = (62 - W) * W = 62W - W^2. To find the maximum area, we need to find the maximum value of this quadratic equation. Since the coefficient of the squared term is negative, the graph of this equation is a downward-opening parabola. The maximum value occurs at the vertex of the parabola. This means the x-coordinate of the vertex, which is -b/2a, will give us the value of W at which the area is maximum. Using the equation W = -b/2a, where a = -1 and b = 62, we find W = 62/2 = 31. Plugging this value back into the equation A = 62W - W^2, we get A = 62 * 31 - 31^2 = 1922 square feet. Therefore, the dimensions of the rectangle with the maximum area are 31 feet by 31 feet.