The formula for the perimeter of a rectangle is 2 * (length + width). To find the dimensions that use the least amount of fencing, we want to minimize the perimeter while still satisfying the constraint that the area is 288 square feet.
So, we can set up the following equation:
2 * (length + width) = minimal perimeter
We know that the area of the rectangle is 288 square feet, so:
length * width = 288
We can substitute the second equation into the first equation to solve for length:
2 * (length + width) = minimal perimeter
2 * length + 2 * width = minimal perimeter
2 * length = minimal perimeter - 2 * width
length = (minimal perimeter - 2 * width) / 2
Substituting this expression for length back into the equation for area:
288 = width * (minimal perimeter - 2 * width) / 2
576 = minimal perimeter * width - 2 * width^2
2 * width^2 + minimal perimeter * width - 576 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
width = (-minimal perimeter + sqrt(minimal perimeter^2 + 4 * 576 * 2)) / (2 * 2)
Since the perimeter is a positive number, the positive solution for width is:
width = (sqrt(minimal perimeter^2 + 4 * 576 * 2) - minimal perimeter) / 4
To find the minimal perimeter, we need to differentiate the expression for width with respect to the perimeter and set the result equal to zero:
d(width)/d(minimal perimeter) = (minimal perimeter / sqrt(minimal perimeter^2 + 4 * 576 * 2) - 1) / 4 = 0
Solving for minimal perimeter, we find:
minimal perimeter = sqrt(576 * 2 * 2) = 24 * sqrt(14)
Finally, we can use this value of the perimeter and the equation for width to find the corresponding length:
length = (minimal perimeter - 2 * width) / 2
= (24 * sqrt(14) - 2 * width) / 2
Thus, the dimensions of the rectangle that use the least amount of fencing are width = sqrt(14) * 6 and length = 12 * sqrt(14).