L = 150 meters
B = 75 meters
Area = 11250 square meters
Step-by-step explanation:
Step 1 :
Total length of fencing = 300 meters
Let L be the length and W be the width of the plot. The river side is not fenced.
Hence we have
L + 2W = 300 => L = 300-2W
Step 2 :
Area of the plot = length * width
=> A = L *W = (300-2W) * W = 300B - 2W²
Step 3:
For the area to be maximum, we need to have the first derivative to be 0 and the second derivative to be less than 0
The first derivative for the area equal to 0 is
300-4W = 0 => 4W = 300 = > W = 75
The second derivative for the area is -4 which is always less than 0.
Hence we have the maximum area when the width is 75 meters.
Step 4 :
When W = 75, the length
L = 300 - 2 * 75 = 300 -150 = 150
So when the length is 150 meters and the width is 75 meters , the area of the plot is maximized
The maximum area that can be enclosed is 75 * 150 = 11250 meters
