Palo Alto College is planning to construct a rectangular parking lot on land bordered on one side by a highway. The plan is to use 560
feet of fencing to fence off the other three sides. What dimensions should the lot have if the enclosed area is to be a maximum?
560/2 = 280
so, the max area is 280x140
divide the fencing equally among lengths and widths.
To find the dimensions of the rectangular parking lot that will maximize the enclosed area, we can use the concept of calculus.
Let's assume that the width of the parking lot is 'w' feet. Since there are three sides of fencing (top, bottom, and one side), the total length of the three sides would be 3w feet.
The remaining side of the parking lot is bordered by a highway, so no fencing will be needed on that side. This means that the length of the lot's fourth side is equal to the width of the highway.
To determine the dimensions that will maximize the enclosed area, we need to express the area (A) of the parking lot in terms of the width (w) of the lot and then find the maximum value of A.
The area of a rectangle is given by the formula A = length × width.
In this case, the length is the width of the highway (let's call it 'h').
So, the area of the parking lot is A = wh.
Since we know that the total length of the three fenced sides is 560 feet, we can express h in terms of w:
h = 560 - 3w.
Substituting this value of h in the area formula, we get:
A = w(560 - 3w).
To find the maximum value of A, we can take the derivative of A with respect to w and find when it is equal to zero.
dA/dw = 560 - 6w.
Setting dA/dw = 0, we get:
560 - 6w = 0.
Solving for w, we find:
w = 560/6 = 93.33 feet.
Therefore, the width of the parking lot that will maximize the enclosed area is approximately 93.33 feet.
We can now substitute this value of w back into the equation for h to find its value:
h = 560 - 3w = 560 - 3(93.33) = 279.99 feet.
Therefore, the dimensions of the rectangular parking lot that will maximize the enclosed area are approximately 93.33 feet (width) by 279.99 feet (length).