A contractor is to fence off a rectangular field along a straight river, the side along the river requiring no fence. What is the greatest area the contractor can fence off using 500m of fencing?

Question

what is the length

Answer 1

Let the length of the rectangular field be L and the width be W.

Since the side along the river requires no fencing, the fencing required is for the other three sides:
- 2 sides of length L
- 1 side of length W

Given that the total fencing available is 500m, we can express this in terms of L and W:
2L + W = 500

We want to maximize the area of the rectangle, which is given by A = LW.

From the previous equation, we can express W in terms of L:
W = 500 - 2L

Substitute this into the equation for the area:
A = L(500 - 2L) = 500L - 2L^2

To find the maximum area, we take the derivative of A with respect to L and set it equal to 0:
dA/dL = 500 - 4L = 0
4L = 500
L = 125

Therefore, the length of the rectangular field should be 125m.

Substitute L = 125 into the equation for W:
W = 500 - 2(125) = 250

So, the width of the rectangular field is 250m.

The greatest area the contractor can fence off is:
A = 125 * 250 = 31250 square meters.