A rectangular field is to be fenced on three sides, leaving a side of 20m uncovered. If the area is 680m2 ,find length of fencing (in m) needed

L*W = 680 m^2.

L*20 = 680, L = 34 m.

2L + W = 2*34 + 20 = 88 m. needed.

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To find the length of fencing needed, we need to calculate the perimeter of the rectangular field.

Given:
Area of the rectangular field: 680m^2
Uncovered side length: 20m

Let's assume the length of the rectangle is L and the width is W.

We know that the formula for the area of a rectangle is A = L * W. Plugging in the given values, we have:

680 = L * W ...(Equation 1)

We are also given that one side of the rectangle is 20m and the remaining three sides need to be fenced. So, the perimeter of the rectangle can be calculated using the formula:

Perimeter = 20 + L + W + L

However, we can replace the value of W from Equation 1:

Perimeter = 20 + L + (680 / L) + L
Perimeter = 20 + 2L + (680 / L)

We are asked to find the length of fencing needed, which is the perimeter. Therefore, we need to solve for L.

To find the value of L, we can differentiate the above equation with respect to L and set it equal to zero. This will give us the maximum or minimum value of L, that is, where the perimeter is minimized:

dP/dL = 2 - ( 680 / L^2 ) = 0 ...(Equation 2)

Simplifying Equation 2, we get:

2 = ( 680 / L^2 )
L^2 = 340
L = √340
L = 18.4 m

Now that we have the value of L, we can substitute it back into Equation 1 to find the value of W:

680 = 18.4 * W
W = 680 / 18.4
W = 37.0 m

Thus, the length of fencing needed (perimeter) is given by:

Perimeter = 20 + L + W + L
Perimeter = 20 + 18.4 + 37.0 + 18.4
Perimeter = 93.8 m

Hence, the length of fencing needed is 93.8 meters.