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 ?
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
To maximize the total area of the three rectangular enclosures, we can use calculus to find the dimensions. Let's denote the width of each enclosure as 'w' and the length of each enclosure as 'l'.
Since the total area of all three enclosures is the sum of their individual areas, we can express the total area as: A = 3wl.
The farmer wants to maximize this area, subject to the constraint that the total length of the enclosures should be equal to the length of the straight river. Let's denote the length of the river as 'L'.
The constraint equation can be written as: L = 3l.
Now, we can express the area in terms of a single variable, 'l', using the constraint equation: A = 3w(L/3) = Lw.
To find the dimensions that maximize the area, we need to differentiate the area equation with respect to the variable 'l' and set it equal to zero, since the maximum or minimum occurs at critical points. So, let's differentiate:
dA/dl = d(Lw)/dl = L(dw/dl) + w(dL/dl).
Since we're maximizing the area, we're looking for a maximum point which corresponds to the derivative being equal to zero. Therefore, we set dA/dl = 0:
0 = L(dw/dl) + w(dL/dl).
Since the enclosures are assumed to be identical, the width 'w' is the same for all three enclosures. Therefore, dw/dl = 0.
Thus, we have 0 = L(0) + w(dL/dl).
0 = w(dL/dl).
From this equation, we observe that w = 0 (which is not a valid solution) or dL/dl = 0.
Since we're assuming that the river is straight, we can conclude that dL/dl = 0. Therefore, the length 'l' of each enclosure should be 0 in order to maximize the total area.
In conclusion, to maximize the total area, the length of each enclosure should be zero, while the width can be any value. This means that the farmer can create three rectangular enclosures with zero length along the straight river.