To maximize the area enclosed by the fencing, Greta should use the remaining three sides to form a rectangle.
Let's denote the length of the garden (parallel to the river) as x and the width of the garden (perpendicular to the river) as y.
Given that there are three sides to fence and the river acts as one side, the total length of fencing used is x + 2y.
Since Greta has 24 yards of fencing material, we have the equation:
x + 2y = 24
To maximize the area A of the rectangle, which is given by A = xy.
Now, let's express one variable in terms of the other. From the fencing equation, we can express x in terms of y:
x = 24 - 2y
Substitute x = 24 - 2y into the area equation:
A = (24 - 2y)y
A = 24y - 2y^2
To find the maximum area, take the derivative of A with respect to y, set it equal to zero, and solve for y:
dA/dy = 24 - 4y
Setting the derivative equal to zero:
24 - 4y = 0
4y = 24
y = 6
Now that we have found y, we can find x using the fencing equation:
x = 24 - 2y = 24 - 2(6) = 24 - 12 = 12
So, the dimensions of the rectangular garden that maximize the area are:
Length (x) = 12 yards
Width (y) = 6 yards
To find the maximum area, substitute these values back into the area equation:
A = 12 * 6 = 72 square yards
Therefore, the maximum area enclosed is 72 square yards.
