A 384 square meter plot of land is to be enclosed by a fence and divided into equal parts by a fence parallel to one pair of sides. What dimensions of the outer rectangle will minimize the amount of force used?
I assume you mean "fence" used.
Three cross fences of length w
area = w L = 384
so w = 384/L
total fence length= t = 3w + 2 L
t = 3*384/L + 2 L
t = 1152/L + 2 L
for min dt/dL = 0
dt/dL = 0 = -1152/L^2 + 2
2 L^2 = 1152
L^2 = 576
L = 24
then w = 16
Do you mean to minimize the amount of FENCE used? I don't see what force has to do with it. It also seeme to me that, by making the outer rectangle very narrow, the amount of fence required to divide it in half can be reduced to zero.
I misread the problem, and forgot that the fence had to also enclose the outer perimter, which was clearly stated.
To find the dimensions of the outer rectangle that will minimize the amount of force used, we need to consider the given information and formulate an equation to represent the force.
Let's assume that the length of the plot of land is "L" meters and the width of the plot of land is "W" meters. Given that the plot of land is a rectangle with an area of 384 square meters, we have the equation:
L * W = 384 (Equation 1)
Since the plot of land needs to be split into equal parts by a fence parallel to one pair of sides, we can divide the plot into two equal rectangles. Let's assume the dimensions of each divided rectangle are L1 and W1:
L1 * W = W1 * L = 384/2 = 192
The fencing required for the outer rectangle has two lengths and two widths:
Fencing = 2L + 2W
We need to make use of the equation for perimeter, which is:
Perimeter = 2L1 + 2W1
To minimize the amount of force used, we need to minimize the perimeter of the divided rectangle.
Using Equation 1, we can substitute W1 = 192 / L1:
Perimeter = 2L1 + 2(192 / L1)
= 2L1 + 384 / L1
To minimize the perimeter, we need to find the minimum value of this expression. We can achieve this by taking the derivative and setting it equal to zero:
dPerimeter / dL1 = 2 - (384 / L1^2) = 0
Simplifying the equation, we have:
2L1^2 - 384 = 0
L1^2 = 384/2
L1^2 = 192
L1 = √192
Now that we have the value of L1, we can substitute it back into the perimeter equation to find the corresponding value for the width, W1:
W1 = 192 / L1
Using the values of L1 and W1, we can calculate the dimensions of the outer rectangle:
Length (L) = 2L1
Width (W) = 2W1
Thus, dimensions of the outer rectangle that will minimize the amount of force used are:
Length (L) = 2 * √192
Width (W) = 2 * (192 / √192)