Jim wants to build a rectangular parking lot along a busy street but only has
1 comma 9001,900
feet of fencing available. If no fencing is required along the street, find the maximum area of the parking lot.
this is just like the river field.
Why do you insist on saying "comma" and then writing the number with the comma? question mark
To find the maximum area of the rectangular parking lot, we need to determine the dimensions of the lot that will use up all of the available fencing.
Let's assume the length of the lot is L and the width is W.
Since the parking lot is located along a busy street, we can assume that the length of the lot is perpendicular to the street, while the width is parallel.
Therefore, we only need to fence the two parallel sides and not the length side, which means we use 2L feet of fencing.
The total length of fencing we have available is given as 1,9001,900 feet. So we can write the equation:
2L + W = 1,9001,900
Now, we need to express the area of the parking lot in terms of L and W. The area, A, of a rectangle is given by the formula:
A = L * W
To find the maximum area, we need to maximize the value of A, while still satisfying the equation 2L + W = 1,9001,900.
We can solve for W by rearranging the equation:
W = 1,9001,900 - 2L
Substituting this value for W in the area formula:
A = L * (1,9001,900 - 2L)
Now, we have the area of the parking lot expressed only in terms of L. To find the maximum area, we can take the derivative of A with respect to L, set it equal to zero, and solve for L. This will give us the critical points where the maximum occurs.
This derivative can be found using calculus:
dA/dL = 1,9001,900 - 4L
Setting dA/dL equal to zero:
1,9001,900 - 4L = 0
Solving for L:
4L = 1,9001,900
L = 1,9001,900 / 4
L = 475,025
Now that we have the value of L, we can substitute it back into the equation W = 1,9001,900 - 2L to find the value of W:
W = 1,9001,900 - 2(475,025)
W = 1,9001,900 - 950,050
W = 950,850
Therefore, the dimensions of the parking lot that will maximize the area are L = 475,025 feet and W = 950,850 feet.
Finally, we can calculate the maximum area by substituting these values into the area formula:
A = L * W
A = 475,025 * 950,850
Therefore, the maximum area of the parking lot is 451,137,738,250 square feet.