Answer:
To find the dimensions that would guarantee the greatest possible area, we can use the formula for the area of a rectangle, which is A = l * w, where l is the length and w is the width.
Since one side of the garden is already boarded by a river, we only need to use 600ft of fencing for the other three sides. This means that the length and width of the garden must add up to 600ft. We can write this as:
l + w = 600
Since we want to find the dimensions that would give us the greatest possible area, we want to maximize l * w. Using the formula for the area of a rectangle, we can write this as:
A = l * w = l * (600 - l) = 600l - l^2
To find the maximum area, we can differentiate this equation and set it equal to zero:
dA/dl = 600 - 2l = 0
Solving for l, we get:
l = 300/2 = 150ft
So, the shorter side is 150ft and the longer side is 450ft (600 - 150).
Finally, we can use the formula for the area of a rectangle to find the greatest possible area:
A = 150 * 450 = 67,500ft^2
Therefore, the greatest possible area for the garden is 67,500ft^2 when the shorter side is 150ft and the longer side is 450ft.
Explanation: