Let $x$ be the length of each side of the square orchid garden in feet. Since the garden is square and the northern side is bounded by the house, only the southern, eastern, and western boundaries require fencing.

Step 1: Determine the total length of fencing needed

• Southern boundary: $x$ feet
• Eastern boundary: $x$ feet
• Western boundary: $x$ feet

The total length of fencing needed is:

$\text{Total fencing} = x + x + x = 3x \text{ feet}$

Step 2: Calculate the cost of fencing

• The southern boundary costs $6 per foot, so the cost of the southern boundary is:

  $\text{Cost of southern boundary} = 6 \times x = 6x$

• The eastern and western boundaries cost $1 per foot each, so the cost for both is:

  $\text{Cost of eastern and western boundaries} = 1 \times x + 1 \times x = 2x$

Step 3: Find the total cost of fencing

The total cost $C(x)$ of the fencing as a function of the side length $x$ is:

$C(x) = 6x + 2x = 8x$

Final Answer

The total cost of the fencing as a function of the length of a side $x$ is:

$\boxed{C(x) = 8x}$

Would you like more details, or do you have any questions?

Related Questions:

1. How would the cost function change if the southern boundary fencing cost increased?
2. What would the total fencing cost be for a garden with a side length of 10 feet?
3. How would the cost function change if the eastern and western boundaries had different costs?
4. What is the cost per foot if the total cost is $240 and the side length is 30 feet?
5. How does the function change if the garden is rectangular rather than square?

Tip:

Always identify each boundary's cost and length separately before combining them to form the total cost function.