Let's break this problem down step-by-step:

Step 1: Set Up the Area Constraint

The area $A$ of the rectangular field is given by: $A = x \cdot y = 13,500,000 \, \text{square feet}$ where:

• $x$ is the length parallel to the dividing fence,
• $y$ is the length perpendicular to the dividing fence.

Since we know the area, we can express $y$ in terms of $x$: $y = \frac{13,500,000}{x}$

Step 2: Express the Fencing Requirement

The total fencing $F$ consists of:

1. Two lengths of $y$ (for the sides perpendicular to the dividing fence),
2. Three lengths of $x$ (two outer lengths plus the dividing fence in the middle).

Thus, the equation for $F$ is: $F(x) = 2y + 3x$ Substitute $y = \frac{13,500,000}{x}$ into this equation: $F(x) = 2 \left( \frac{13,500,000}{x} \right) + 3x$ Simplify: $F(x) = \frac{27,000,000}{x} + 3x$

Step 3: Find the Derivative $F'(x)$

To find the derivative $F'(x)$, use the power rule: $F(x) = \frac{27,000,000}{x} + 3x = 27,000,000 \cdot x^{-1} + 3x$ Differentiating with respect to $x$: $F'(x) = -\frac{27,000,000}{x^2} + 3$

Step 4: Find the Critical Numbers

To find the critical points, set $F'(x) = 0$: $-\frac{27,000,000}{x^2} + 3 = 0$ Solving for $x$: $\frac{27,000,000}{x^2} = 3$ $x^2 = \frac{27,000,000}{3} = 9,000,000$ $x = \sqrt{9,000,000} = 3000$

So, the critical number is: $x = 3000$

Summary of Answers

1. The equation for $F(x)$ in terms of $x$ is: $F(x) = \frac{27,000,000}{x} + 3x$
2. The derivative $F'(x)$ is: $F'(x) = -\frac{27,000,000}{x^2} + 3$
3. The critical number is: $x = 3000$

Would you like more details on any step?

Here are some follow-up questions for further exploration:

1. What does the critical number $x = 3000$ represent in terms of the fencing optimization?
2. How would the function $F(x)$ change if the area requirement increased or decreased?
3. Can we confirm that the critical point minimizes or maximizes the fencing length?
4. How would the solution change if the dividing fence ran perpendicular to $x$ instead?
5. What are the implications for practical fencing in terms of materials and cost based on $F(x)$?

Tip: Always double-check each step when setting up an equation to ensure no details are missed!