Final answer:
To find the maximum area of the rectangular parking lot with 1,900 feet of fencing, we need to write equations for the length and width of the parking lot. By using the formula for the vertex of a quadratic equation, we find that the maximum area is 451,250 square feet.
Step-by-step explanation:
To find the maximum area of the rectangular parking lot, we need to use the given information about the fencing. Let's represent the length of the parking lot as L and the width as W. We know that the total amount of fencing is 1,900 feet, and since no fencing is required along the street, the total length of the fencing will be used for the other three sides of the rectangle. This means that 2L + W = 1,900.
Since we want to maximize the area of the parking lot, we can express the area as A = L * W. To solve for the maximum area, we can rewrite the equation for the total length of the fencing as W = 1,900 - 2L. Substituting this into the area equation, we have A = L * (1,900 - 2L). This is a quadratic equation, and to find the maximum area, we can find the vertex of the parabola formed by this equation.
The x-coordinate of the vertex can be found using the formula x = -b/2a. In this case, a = -2, b = 1,900. Substituting these values, we get L = -1,900 / (2*-2) = 1,900 / 4 = 475 feet. Substituting this value back into the equation for W, we find W = 1,900 - 2(475) = 950 feet.
Therefore, the maximum area of the parking lot is A = L * W = 475 * 950 = 451,250 square feet.