From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $20 per foot and the fence for the middle costs $8 per foot. If each lot contains 15,360 square feet, find the dimensions of each lot that yield the minimum cost for the fence.
length of side parallel to the middle fence	
135
 
Incorrect: Your answer is incorrect.
length of side perpendicular to the middle fence

To find the dimensions of each lot that minimize the cost of the fence, let's set up the problem.

Let:
- $ x $ = length of the side parallel to the middle fence (the longer side).
- $ y $ = length of the side perpendicular to the middle fence (the shorter side).

Since each lot contains $ 15,360 $ square feet, we have:
\[
x \times y = 15,360
\]
From this, we can express $ y $ in terms of $ x $:
\[
y = \frac{15,360}{x}
\]

### Cost Calculation
The cost of the fence consists of two components:
1. The perimeter of the outer fence, which is:
\[
\text{Perimeter} = 2x + 2y
\]
2. The cost of the middle fence, which is:
\[
\text{Middle Fence} = y
\]

### Total Cost
The total cost $ C $ can be expressed as:
\[
C = 20(2x + 2y) + 8y = 40x + 40y + 8y = 40x + 48y
\]
Substituting $ y $:
\[
C = 40x + 48 \left(\frac{15,360}{x}\right)
\]
\[
C = 40x + \frac{734,080}{x}
\]

### Finding the Minimum Cost
To minimize the cost, we take the derivative of $ C $ with respect to $ x $ and set it to zero:
\[
C' = 40 - \frac{734,080}{x^2}
\]
Setting the derivative equal to zero gives:
\[
40 = \frac{734,080}{x^2}
\]
\[
x^2 = \frac{734,080}{40} = 18,352
\]
\[
x = \sqrt{18,352} \approx 135.49 \text{ feet}
\]

### Finding $ y $
Now, substitute $ x $ back to find $ y $:
\[
y = \frac{15,360}{135.49} \approx 113.5 \text{ feet}
\]

### Conclusion
The approximate dimensions of each lot that yield the minimum cost for the fence are:
- Length of the side parallel to the middle fence ($ x $) ≈ **135.49 feet**
- Length of the side perpendicular to the middle fence ($ y $) ≈ **113.5 feet**

Do you need any further details or clarifications? Here are some related questions:

1. What is the total cost of the fence with these dimensions?
2. How would the cost change if the price per foot of the outer fence increases?
3. What are the implications of changing the area of each lot?
4. How would the dimensions change if the cost of the middle fence was higher?
5. Can this problem be applied to different shapes or configurations of fencing? 

**Tip:** Always verify the dimensions with the given area constraint to ensure accuracy.

From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $20 per foot and the fence for the middle costs $8 per foot. If each lot contains 15,360 square feet, find the dimensions of each lot that yield the minimum cost for the fence.

Explore the problem of minimizing fencing costs for rectangular lots. Given that each lot is 15,360 square feet, this solution includes a detailed calculation of optimal dimensions and cost breakdowns. Suitable for high school students studying optimization and calculus.

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