a field bounded on one side by a river is to be fenced on three sides to form a rectangular enclosure. there are 320 ft of fencing available. what should the dimensions be to have an enclosure with the maximum possible area?
The largest area would be a square.
360/3 = 120
To find the dimensions that give the maximum possible area for the rectangular enclosure, we need to use the fact that the field is bounded on one side by a river and fenced on the other three sides. The total amount of fencing available is 320 ft.
Let's denote the length of the enclosure as "L" and the width as "W".
To maximize the area of the enclosure, we should aim to make it as close to a square as possible. This is because a square has the largest area among all rectangles with the same perimeter.
Since we have a river on one side, we only need to fence three sides. Therefore, the total length of fencing required is the sum of one length (L), one width (W), and two widths (2W) to enclose the remaining three sides.
So, the total length of fencing can be expressed as: L + W + 2W = L + 3W = 320 ft.
Now, we need to express one variable in terms of the other to create an equation that will allow us to solve for the dimensions. Let's isolate L:
L = 320 ft - 3W.
Substituting this back into the area formula for a rectangle:
Area = L * W = (320 ft - 3W) * W = 320W - 3W^2.
To maximize the area, we need to find the value of W that maximizes the equation.
To do this, we can take the derivative of the area equation with respect to W and set it equal to zero:
d(Area)/dW = 320 - 6W = 0.
Solving this equation gives:
6W = 320,
W = 320 / 6,
W = 53.33 ft.
Now, substituting this value of W back into the length equation:
L = 320 ft - 3(53.33 ft),
L = 320 ft - 159.99 ft,
L = 160.01 ft.
So, the dimensions that will result in the maximum possible area for the rectangular enclosure are approximately 53.33 ft by 160.01 ft.