What are the dimensions of a rectangular field of area A that requires the least amount of fencing.
see my previous reply to this same idea.
Reiny, may i have a solution for this?
To find the dimensions of the rectangular field with the least amount of fencing for a given area A, you need to consider the relation between the dimensions of the field and its perimeter. Let's denote the length of the field as L and the width as W.
The formula for the perimeter (P) of a rectangle is given by:
P = 2L + 2W
Since we are looking for the minimum amount of fencing, we need to minimize the perimeter. Given the constraint that the area (A) remains fixed, we have the equation:
A = L * W
To find the dimensions that minimize the perimeter, we can solve for one variable in terms of the other and then substitute it into the perimeter equation.
First, solve the area equation for L:
L = A / W
Substitute this equation into the perimeter formula:
P = 2(A / W) + 2W
To minimize P, we need to find the value of W that makes the derivative of P with respect to W equal to zero. Take the derivative of P:
dP/dW = -2A/W^2 + 2
Set dP/dW equal to zero and solve for W:
-2A/W^2 + 2 = 0
Simplifying this equation gives:
2A = 2W^2
W^2 = A
W = √A
By substituting this value of W back into the equation for L, we can find the dimensions of the field:
L = A / √A
Thus, the dimensions of the rectangular field with the least amount of fencing for a given area A are L = A / √A and W = √A.