Let's solve the problem step by step.

Step 1: Define Variables

Let:

• $x$ be the length of the rectangular plot (in feet),
• $y$ be the width of the rectangular plot (in feet).

Step 2: Cost Equation

We are given that:

• The two opposite sides of the rectangle will use heavy-duty fencing, costing $5 per foot.
• The other two opposite sides will use standard fencing, costing $2 per foot.

The total cost of fencing is $11,000. The cost for the heavy-duty fencing will be for the two sides of length $x$, and the cost for the standard fencing will be for the two sides of length $y$.

Thus, the cost equation is: $5(2x) + 2(2y) = 11000$ Simplifying: $10x + 4y = 11000$

Step 3: Area Equation

The area of the rectangular plot is given by: $A = x \cdot y$ We want to maximize this area.

Step 4: Solve the Cost Equation for One Variable

From the cost equation: $10x + 4y = 11000$ Solve for $y$ in terms of $x$: $4y = 11000 - 10x$ $y = \frac{11000 - 10x}{4}$ $y = 2750 - 2.5x$

Step 5: Substitute $y$ into the Area Formula

Substitute $y = 2750 - 2.5x$ into the area formula $A = x \cdot y$: $A(x) = x(2750 - 2.5x)$ $A(x) = 2750x - 2.5x^2$

Step 6: Maximize the Area

To maximize the area, we take the derivative of $A(x)$ with respect to $x$ and set it equal to 0: $\frac{dA}{dx} = 2750 - 5x$ Set the derivative equal to 0: $2750 - 5x = 0$ Solving for $x$: $x = \frac{2750}{5} = 550$

Step 7: Find $y$

Substitute $x = 550$ into the equation for $y$: $y = 2750 - 2.5(550)$ $y = 2750 - 1375 = 1375$

Step 8: Verify the Cost

Now, verify that the total cost is $11,000:

• The cost of the two heavy-duty sides is $10x = 10(550) = 5500$ dollars.
• The cost of the two standard sides is $4y = 4(1375) = 5500$ dollars.

Total cost: $5500 + 5500 = 11000 \, \text{dollars}$ This matches the given cost.

Final Answer:

• Heavy-duty fencing: 1100 feet (since $2x = 2(550)$)
• Standard fencing: 2750 feet (since $2y = 2(1375)$)

Would you like more details or further explanations? Here are some related questions:

1. How would the area change if we used only heavy-duty fencing for all four sides?
2. What happens if the cost of heavy-duty fencing increases to $6 per foot?
3. How do we calculate the maximum area if the total cost of fencing is reduced to $8000?
4. What is the formula to calculate the total cost based on different lengths of fencing?
5. How do you find the dimensions of a rectangle that maximizes area given a fixed perimeter?

Tip: When optimizing a function like the area, the derivative helps to find the critical points where the maximum or minimum might occur. Always check the second derivative to confirm if it’s a maximum!