A farmer wants to fence an area of 6 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?
ft (smaller value)
ft (larger value)
I got 2000 ft for smaller value and 3000 ft for larger value. Just wanna verify with someone else if they agree or disagree with my answer
That is correct. Good job.
As usual in problems like this, maximum area or minimum fencing is achieved when the total fence is divided equally between lengths and widths.
To minimize the cost of the fence, we need to find the dimensions that minimize the amount of fencing required. Let's assume the sides of the rectangular field are x and y.
Given that the area of the rectangular field is 6 million square feet, we have xy = 6,000,000.
We also know that the field will be divided into two equal halves with a fence parallel to one of the sides. Therefore, one side of the rectangle will be x, and the other side will be y/2.
The total amount of fencing required is given by the perimeter of the rectangular field, which can be calculated as:
P = 2x + 2(y/2) = 2x + y
To minimize the cost, we need to minimize the amount of fencing required, which means minimizing the perimeter. In this case, we want to minimize the function P = 2x + y.
To find the values of x and y that minimize P, we can use calculus. We need to find the critical points where the partial derivatives of P with respect to x and y are equal to zero.
∂P/∂x = 2
∂P/∂y = 1
Setting these partial derivatives equal to zero, we get:
2 = 0
1 = 0
These equations have no solutions, which means there are no critical points.
Therefore, we can conclude that the minimum perimeter (and thus, the minimum amount of fencing required) occurs when the sides of the rectangular field are as close to each other as possible, which means x and y should be as close to the square root of 6,000,000 as possible.
Calculating the square root of 6,000,000 gives us approximately:
sqrt(6,000,000) ≈ 2,449.49
So, the sides of the rectangular field should be approximately 2,449.49 ft (smaller value) and 4,898.98 ft (larger value).
To minimize the cost of the fence, we need to find the dimensions of the rectangular field that will minimize the amount of fencing material required.
Let's assume the length of the rectangular field is 'x' feet and the width is 'y' feet. Since the area of the rectangular field is given as 6 million square feet, we can write the equation:
xy = 6,000,000
Now we want to divide the field into two equal halves with a fence parallel to one of the sides. This means we need a fence running along the width (y-axis) of the field. So, we can write another equation for the total length of the fence required:
Total Fence Length = 2x + y
We want to minimize this fence length equation, so we can express y in terms of x using the area equation:
y = 6,000,000 / x
Substituting this value of y in the fence length equation, we get:
Total Fence Length = 2x + (6,000,000 / x)
Now, we can minimize this equation by taking the derivative with respect to x and setting it equal to zero:
d(Total Fence Length) / dx = 2 - (6,000,000 / x^2) = 0
Simplifying further:
2 = 6,000,000 / x^2
x^2 = 6,000,000 / 2
x^2 = 3,000,000
Taking the square root of both sides:
x = sqrt(3,000,000)
x ≈ 1732 feet
Now, we can substitute this value of x back into the area equation to find y:
y = 6,000,000 / x
y = 6,000,000 / 1732
y ≈ 3463 feet
Therefore, the dimensions of the rectangular field that will minimize the cost of the fence are approximately:
1732 ft (smaller value)
3463 ft (larger value)