Jim wants to build a rectangular parking lot along a busy street but only has 1,900
feet of fencing available. If no fencing is required along the street, find the maximum area of the parking lot.
Hey, I just did one like this for you.
451,250
To find the maximum area of the rectangular parking lot, we need to determine the dimensions of the parking lot that will use up all the fencing available (1,900 feet) while maximizing the area.
Let's assume the length of the parking lot is x feet. Since there is no fencing required along the street, the width of the parking lot will be the remaining fencing, which is 1,900 - 2x feet (since there are two sides of length x).
The area of a rectangle is calculated by multiplying the length by the width. So, the area (A) in terms of x is:
A = x * (1900 - 2x)
We want to maximize the area, so we need to find the value of x that gives us the maximum value of A. To do this, we can find the vertex of the quadratic function A.
First, let's expand the expression:
A = 1900x - 2x^2
To find the vertex, we can use the formula x = -b/2a. In this case, a = -2 and b = 1900. Substituting these values into the formula:
x = -1900 / (2 * -2)
x = -1900 / -4
x = 475
The length of the parking lot that maximizes the area is 475 feet. To find the width, we can substitute this value back into the expression for the width:
Width = 1900 - 2x
Width = 1900 - 2 * 475
Width = 950 feet
Therefore, the maximum area of the parking lot is:
A = length * width
A = 475 * 950
A = 451,250 square feet