Let's define the variables and the setup:

• The vegetable patch is square, so each side of the patch has the same length $x$ feet.
• The cost of fencing for the east and west sides is $6 per foot.
• The cost of fencing for the north and south sides is $5 per foot.

Step-by-step process:

1. The east and west sides have a combined length of $2x$ feet. Since the cost is $6 per foot for these two sides, the total cost for the east and west sides is: $\text{Cost for east and west sides} = 2x \times 6 = 12x$

2. The north and south sides also have a combined length of $2x$ feet. The cost for these sides is $5 per foot, so the total cost for the north and south sides is: $\text{Cost for north and south sides} = 2x \times 5 = 10x$

3. The total cost $C(x)$ of fencing all four sides is the sum of the costs for the east and west sides and the north and south sides: $C(x) = 12x + 10x = 22x$

Final Answer:

The total cost of the fencing as a function of the length $x$ (in feet) is: $C(x) = 22x \, \text{dollars}$

Would you like further details, or have any questions?

Related Questions:

1. What would the total cost be if the side length $x$ is 20 feet?
2. How would the cost function change if the prices per foot increased by $1 for each side?
3. If you wanted to double the fenced area, how would the side length change, and what would be the new cost?
4. What is the total length of fencing required for a side length $x = 15$ feet?
5. If the east and west sides had different prices for each, how would the cost function change?

Tip:

When working with cost functions, always keep track of units (feet, dollars, etc.) to ensure accuracy.