Final answer:
The dimensions that maximize the enclosed area, solve the system of equations formed by the perimeter equation and the area equation, and find the maximum value of the quadratic function. Plug in the value of x into the perimeter equation to solve for y and obtain the dimensions that maximize the enclosed area.
Step-by-step explanation:
To find the dimensions that maximize the enclosed area, we can use the formula for the area of a rectangle: Area = length x width.
Let's call the length of each rectangle x and the width y. The total perimeter of the rectangles is 2300 feet, which can be expressed as 2x + 3y = 2300. We can solve this system of equations to find the values of x and y that maximize the area.
Step 1: Rewrite the equation 2x + 3y = 2300 in terms of y: y = (2300 - 2x) / 3.
Step 2: Substitute the value of y into the area formula: Area = x * ((2300 - 2x) / 3). Simplify the expression to get a quadratic function: Area = (2300/3)x - (2/3)x^2.
Step 3: To find the dimensions that maximize the area, we need to find the maximum value of the quadratic function. The maximum value occurs at the vertex of the parabola, which is the x-coordinate of the vertex. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -2/3 and b = 2300/3. Plug in these values to find x.
Step 4: Once you have the value of x, substitute it back into the equation 2x + 3y = 2300 to solve for y. This will give you the dimensions x and y that maximize the enclosed area.