Suppose a rectangular pasture is to be constructed using [tex]\frac{3}{4}[/tex] linear mile of fencing. The pasture will have two dividers parallel to one pair of sides and three dividers parallel to the other pair of sides, so that there are twelve congruent enclosures. What is the maximum total area of such a pasture, in square feet?

To maximize the total area, divide the fencing into twelve congruent enclosures by using two parallel dividers on one pair of sides and three dividers on the other pair of sides.

Explanation:

To maximize the total area of the pasture, we need to divide the fencing into twelve congruent enclosures. Let's assume the length of the rectangular pasture is x miles and the width is y miles. We have a total of 3 + 4 + 4 = 11 miles of fencing to use.

Since we need two parallel dividers on one pair of sides and three dividers on the other pair of sides, we can calculate the length and width of each enclosure by dividing the length and width of the pasture into equal segments.

The length of each enclosure is (x-2)/3 miles and the width is (y-2)/4 miles. The total area of each enclosure is (x-2)/3 * (y-2)/4 square miles. Finally, to get the maximum total area, we can multiply the area of each enclosure by twelve: 12 * (x-2)/3 * (y-2)/4 square miles.

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