Let the width of each enclosure be x and the length be y.
Therefore, the total area of the three enclosures is 3xy.
Since they are to be identical, we can consider one enclosure and then multiply the area by 3.
The farmer wants to make the enclosures along a straight river, which means that one of the sides of each enclosure is already fenced by the river.
Let's assume that the river is the length of the rectangle and that the farmer has a fixed amount of fencing material for the remaining sides.
Therefore, the perimeter (fenced sides) of the rectangle is 2x + y.
Since the farmer has a fixed amount of fence material, we can write the equation:
2x + y = k, where k is a constant
Solving for y, we get:
y = k - 2x
Now, we can substitute y into the area equation:
A = xy
A = x(k - 2x)
A = -2x^2 + kx
To find the maximum area, we take the derivative of A with respect to x and set it equal to zero:
dA/dx = -4x + k = 0
Solving for x, we get:
x = k/4
Substituting this value of x back into the equation for y, we get:
y = k/2
Therefore, the dimensions of the enclosure with maximum area are:
width = x = k/4
length = y = k/2
Since there are three identical enclosures, the farmer will need:
Total width = 3x = 3k/4
Total length = y = k/2
Therefore, the dimensions of each enclosure are:
width = k/4
length = k/2
A farmer wants to make three identical rectangular enclosures (same width and same length ) along a straight river, need not fence, What should be the dimensions (length and width ) of each enclosure if the total area of the enclosure is to be maximum ?
1 answer