Final answer:
The equation that can be used to calculate the area enclosed by the fencing is A(w) = -w^2 + 132w, and the dimensions that maximize the rectangular area are 66 feet by 66 feet.
Step-by-step explanation:
To calculate the area enclosed by the fencing, we need to determine the dimensions of the rectangular area. Let's assume that the width of the rectangle is 'w' feet. Since we have 264 feet of fencing, we can write the equation 2w + 2L = 264, where L represents the length of the rectangle. Simplifying the equation, we get 2w + 2L = 264. Solving for L, we find L = 132 - w.
To find the equation for the area enclosed by the fencing, we use the formula A = w * L. Substituting the value of L, we get A = w * (132 - w). This simplifies to A = -w^2 + 132w.
In order to maximize the rectangular area, we need to find the maximum point of the quadratic equation A(w) = -w^2 + 132w. The maximum occurs when the coefficient of the quadratic term is negative, which is the case here. The maximum value is achieved when w = -b/2a, where the equation is in the form A(w) = aw^2 + bw + c. Plugging in the values a = -1 and b = 132, we find w = 66.
Therefore, the equation that can be used to calculate the area enclosed by the fencing is A(w) = -w^2 + 132w, and the dimensions that maximize the rectangular area are 66 feet by 66 feet.