Answer:
To find the dimensions that will guarantee the greatest possible area for the rectangular garden, we can use the information given and apply some mathematical reasoning.
Let's denote the length of the garden as L and the width as W. According to the question, one side of the garden is bordered by the river, so it does not require any fencing. This means that only three sides of the garden need to be fenced.
We know that the gardener has 920 feet of fencing available. Since there are three sides to fence, we can set up the equation:
2L + W = 920
We need to find the dimensions that maximize the area of the garden, which is given by the formula A = L * W.
To proceed, we can solve the equation for one of the variables and substitute it into the area formula. Let's solve the equation for W:
W = 920 - 2L
Now we substitute this value of W into the area formula:
A = L * (920 - 2L)
To find the maximum area, we need to find the value of L that maximizes the area. We can do this by finding the vertex of the quadratic equation A = -2L^2 + 920L.
The vertex of a quadratic equation in the form A = aL^2 + bL + c is given by L = -b / (2a). In our case, a = -2 and b = 920.
Substituting these values, we find:
L = -920 / (2*(-2))
L = 920 / 4
L = 230
So, the length of the garden that will guarantee the greatest possible area is 230 feet.
To find the width, we can substitute this value of L into our equation for W:
W = 920 - 2L
W = 920 - 2(230)
W = 460
Therefore, the width of the garden that will guarantee the greatest possible area is 460 feet.
In conclusion, the dimensions that will guarantee the greatest possible area for the garden are a length of 230 feet and a width of 460 feet.