A rectangular lot having an area of
5000 m^2 is to be fenced on sides. To have the least moment of fencing , what should be the ratio of its length to its width?
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To minimize the moment of fencing for a rectangular lot, we need to find the dimensions that minimize the perimeter of the fence.
Let's assume the length of the rectangle is L and the width is W.
The area of the rectangle is given as 5000 m^2, so we have the equation:
L * W = 5000
The perimeter of the fence, which we want to minimize, is given by:
P = 2L + 2W
We can solve for either L or W in terms of the other variable using the area equation:
L = 5000 / W
Substituting this expression for L into the perimeter equation, we have:
P = 2 * (5000 / W) + 2W
To find the minimum value of P, we can take the derivative with respect to W and set it equal to 0:
dP/dW = -10000 / W^2 + 2 = 0
Simplifying this equation, we have:
-10000 / W^2 + 2 = 0
-10000 = 2W^2
W^2 = -5000
This is not possible, so there is no minimum value for the perimeter. Therefore, we cannot find a ratio of length to width that minimizes the moment of fencing for a rectangular lot with an area of 5000 m^2.
To determine the ratio of the length to the width that will result in the least moment of fencing, we need to take into account the formula for the moment of a force, which is given by the product of the force and its perpendicular distance from a reference point.
In this case, the force is the length of the fencing, and its perpendicular distance is the width of the rectangular lot.
Let's assume:
Length of the rectangular lot = L (in meters)
Width of the rectangular lot = W (in meters)
We are given that the area of the rectangular lot is 5000 m^2, so we have:
L * W = 5000
To minimize the moment of the fencing, we need to minimize the value of L * W while satisfying the above equation.
Now, let's solve for L and express it in terms of W:
L = 5000 / W
Substituting this back into the formula for the moment of the fencing:
Moment = L * W
Moment = (5000 / W) * W
Moment = 5000
As we can see, the value of the moment is independent of the width, W. Therefore, the moment of the fencing will be the same regardless of the ratio of the length to the width.
In conclusion, the ratio of the length to the width does not affect the least moment of the fencing and can vary depending on other factors such as the shape and design preferences.