Let's break down the total cost of fencing as a function of the length of a side $x$ (in feet) for the square vegetable patch.

1. Since it's a square, each side has the same length $x$.

2. The east and west sides (2 sides) cost $7 per foot. So, the total cost for the east and west sides is: $2 \cdot 7 \cdot x = 14x$

3. The north and south sides (2 sides) cost $5 per foot. So, the total cost for the north and south sides is: $2 \cdot 5 \cdot x = 10x$

4. Therefore, the total cost $C(x)$ of fencing the entire patch is the sum of the costs for all four sides: $C(x) = 14x + 10x = 24x$

Thus, the total cost of the fencing as a function of the side length $x$ is: $C(x) = 24x$

Do you have any questions or need further details?

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Here are 5 related questions:

1. If the side length of the patch is 10 feet, what is the total cost of the fencing?
2. How much would it cost if the fencing prices changed to $8 for east and west, and $6 for north and south?
3. How would the cost change if the patch was rectangular instead of square?
4. What if you wanted to fence in two patches side by side, each with the same dimensions?
5. Can we derive the cost function if the prices per foot increase by a fixed percentage each year?

Tip: Always double-check unit consistency when setting up equations involving costs or dimensions.