A farmer wishes to make two rectangular enclosures with no fence along the river and a 10m opening for a tractor to enter. If 1034 m of fence is available, what will the dimension of each enclosure be for their areas to be a maximum?

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43 To find the dimensions of each enclosure for their areas to be a maximum, we need to use optimization techniques.

Let's denote the length of one enclosure as L, and the length of the other enclosure as W. Since there is no fence along the river, L will be the length of the enclosure perpendicular to the river, and W will be the length of the enclosure parallel to the river.

Given that there is a 10m opening for the tractor to enter, we can determine the total length of the fence required as:

2L + 3W = 1034

Now, we need to express the total area of the two enclosures in terms of L and W:

Total Area = Area of First Enclosure + Area of Second Enclosure
= L * W + L * (W - 10)
= LW + LW - 10L
= 2LW - 10L

To maximize the total area, we need to find the critical points by taking the derivative of the area function with respect to L and setting it equal to zero:

d/dL (2LW - 10L) = 2W - 10 = 0

Solving this equation, we get:

2W = 10
W = 5

Substituting this value back into the equation 2L + 3W = 1034, we can solve for L:

2L + 3(5) = 1034
2L + 15 = 1034
2L = 1034 - 15
2L = 1019
L = 1019 / 2
L = 509.5

Therefore, the dimensions of each enclosure for their areas to be a maximum are approximately 509.5m by 5m.