A farmer has 320 meters of fencing wire. He wants to enclose a rectangular garden one side of which is a river. Find the dimensions of the maximum area possible he can fence if he does not need to fence on the side where there is a river.
Let the length of the garden be represented by x meters. Therefore, the width of the garden will be (320 - x)/2 meters.
The area of the garden is given by A = length x width
A = x*(320 - x)/2
A = 160x - x^2/2
To find the maximum area, we need to find the derivative of the area with respect to x and set it equal to 0.
dA/dx = 160 - x = 0
x = 160
Therefore, the dimensions of the maximum area possible are:
Length = 160 meters
Width = (320 - 160)/2 = 80 meters
Therefore, the farmer should enclose a rectangular garden with dimensions 160 meters by 80 meters to maximize the area.