A farmer wants to fence off a rectangular field of area 15000 square feet using barbed wire fencing. SInce the opposite of the road is a corn-field, he wants a two-strand fence along the road, and one strand on each of the other three sides. What are the dimensions of the field which minimize the amount of fence used?
if the 2-strand dimension is x, then we have
f(x) = 2x + x + 2(15000/x)
df/dx = 3 - 30000/x^2
min fence when df/dx = 0, at x = 100
so, the field is 100x150
as usual, the fencing is divided equally among the dimensions.
f(x) = fencing(x) !
To minimize the amount of fence used, we can start by determining the dimensions of the rectangular field.
Let's assume the length of the field is L and the width is W.
The area of the rectangular field is given as 15000 square feet.
Therefore, we have the equation: L * W = 15000.
To minimize the amount of fence used, we need to minimize the perimeter of the rectangular field.
The perimeter is given by: 2L + 2W.
Since there will be two strands of fence along the road (which is the longer side), the length of the road side will be twice the length of the other sides.
The perimeter equation can now be modified as: L + 2W.
Next, we need to express one variable in terms of the other.
We can solve the area equation L * W = 15000 for L:
L = 15000 / W.
Substituting this into the modified perimeter equation, we get:
(15000 / W) + 2W.
To minimize the amount of fence used, we need to find the critical points by taking the derivative and setting it equal to zero.
Let's find the derivative of the expression:
d/dW [(15000 / W) + 2W] = -15000 / W^2 + 2.
Setting this equal to zero:
-15000 / W^2 + 2 = 0.
Simplifying the equation, we get:
-15000 = -2W^2.
Dividing both sides by -2:
W^2 = 7500.
Taking the square root of both sides:
W = sqrt(7500) ≈ 86.60.
Since the dimensions of a rectangular field cannot be negative, we can disregard the negative root.
Now that we have the value of W, we can find the value of L by substituting it back into the area equation:
L = 15000 / W = 15000 / 86.60 ≈ 173.21.
Therefore, the dimensions of the field that minimize the amount of fence used are approximately 173.21 feet by 86.60 feet.
To find the dimensions of the field that minimize the amount of fence used, we need to determine the dimensions of the rectangle.
Let's consider the length of the field to be 'L' and the width to be 'W'.
Since there are two strands of fence along the road and one strand on each of the other three sides, the total length of the fencing required is given by the perimeter of the rectangle.
Perimeter (P) = 2L + W
We also know that the area (A) of the rectangle is given by:
Area (A) = L * W
Given that the area of the field is 15000 square feet, we have:
L * W = 15000 ... Equation (1)
We want to minimize the amount of fencing used, which is the perimeter.
To proceed, let's solve Equation (1) for L:
L = 15000 / W
Substitute this expression for L in the formula for the perimeter:
P = 2L + W
= 2(15000 / W) + W
Now, we can find the derivative of P with respect to W and set it equal to zero to find the critical points.
dP/dW = -30000 / W^2 + 1 = 0
Simplifying, we get:
30000 = W^2
Taking the square root of both sides, we find:
W = sqrt(30000)
W ≈ 173.21
So, the width of the field is approximately 173.21 feet.
Substituting this value of W back into Equation (1), we can find the corresponding length:
L = 15000 / W
L ≈ 15000 / 173.21
L ≈ 86.61
Therefore, the dimensions of the field that minimize the amount of fence used are approximately 86.61 feet by 173.21 feet.