To solve this problem, let's start by breaking down the information provided:

1. The total area of the rectangular field is $3,000,000$ square feet.
2. The field will be divided in half with a fence down the middle, parallel to one of its sides.
3. We want to minimize the total length of the fencing used.

Step-by-Step Solution

Step 1: Define Variables

Let:

• $x$ = length of the field
• $y$ = width of the field

Since the area $A$ of a rectangle is given by: $A = x \times y$ we have: $x \times y = 3,000,000$

Step 2: Divide the Field in Half

The rancher divides the field with an additional fence parallel to the shorter side. This implies there will be two lengths and three widths of fencing.

Step 3: Express the Total Fencing Length

The total length of the fencing $L$ required is: $L = 2x + 3y$

Step 4: Substitute $y$ in Terms of $x$

From $x \times y = 3,000,000$, we can solve for $y$: $y = \frac{3,000,000}{x}$

Substituting this into the fencing equation: $L = 2x + 3\left(\frac{3,000,000}{x}\right)$ $L = 2x + \frac{9,000,000}{x}$

Step 5: Minimize the Total Fencing Length

To find the minimum $L$, take the derivative of $L$ with respect to $x$ and set it to zero.

$\frac{dL}{dx} = 2 - \frac{9,000,000}{x^2} = 0$

Solving for $x$: $2 = \frac{9,000,000}{x^2}$ $x^2 = \frac{9,000,000}{2}$ $x = \sqrt{4,500,000}$ $x \approx 2121.32 \text{ feet}$

Step 6: Calculate $y$

Using $y = \frac{3,000,000}{x}$: $y = \frac{3,000,000}{2121.32} \approx 1414.21 \text{ feet}$

Step 7: Calculate the Minimum Length of Fencing

Now, substitute $x \approx 2121.32$ and $y \approx 1414.21$ into $L = 2x + 3y$: $L = 2(2121.32) + 3(1414.21)$ $L \approx 4242.64 + 4242.63$ $L \approx 8485.27 \text{ feet}$

Answer

The shortest length of fence the rancher can use is approximately 8485.27 feet.

Would you like more details or have any questions?

Here are some related questions to expand on this problem:

1. How would the solution change if the field were divided into four equal parts instead?
2. What if the area were different but the division in half remained the same?
3. Could we solve this using calculus techniques for constrained optimization, such as the Lagrange multiplier?
4. How would the solution differ if the fence were only required around the perimeter?
5. How does the shape of a rectangle affect the fencing requirement for a given area?

Tip: When optimizing, always express all variables in terms of a single variable before taking derivatives; this simplifies the calculation.