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?
what is the length
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.