A farmer wants to fence an area of 13.5 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. Let y represent the length (in feet) of a side perpendicular to the dividing fence, and let x represent the length (in feet) of a side parallel to the dividing fence.
Let F represent the length of fencing in feet. Write an equation that represents F in terms of the variable x.
F(x) = 
  
Incorrect: Your answer is incorrect.
Find the derivative F ′(x).
F ′(x) = 
  
Incorrect: Your answer is incorrect.
Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
x = 
  
Incorrect: Your answer is incorrect.

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!

A farmer wants to fence an area of 13.5 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. Let y represent the length (in feet) of a side perpendicular to the dividing fence, and let x represent the length (in feet) of a side parallel to the dividing fence. Let F represent the length of fencing in feet. Write an equation that represents F in terms of the variable x.
Find the derivative F ′(x).
Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Explore how to solve an optimization problem involving a 13.5 million square feet rectangular field, calculating the fencing required for dividing it in half. Includes step-by-step setup of the fencing equation, derivative calculation, and critical number analysis for minimizing fencing. Ideal for Grade 11-12 calculus students.

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