To determine how much fencing Kai will need for all four sides of his garden, we first need to ascertain the dimensions of the garden based on the area of 121 square feet.
Assuming the garden is rectangular, we can use the formula for area:
\[ \text{Area} = \text{length} \times \text{width} \]
Let’s denote the length as \(L\) and the width as \(W\). From the area given:
\[ L \times W = 121 , \text{ft}^2 \]
Next, to find the perimeter, which is the total length of fencing needed, we use the formula for perimeter \(P\) of a rectangle:
\[ P = 2L + 2W = 2(L + W) \]
However, we do not have specific measurements for \(L\) and \(W\). We can express the perimeter in terms of one variable. If we express \(W\) in terms of \(L\):
\[ W = \frac{121}{L} \]
Substituting back into the perimeter formula gives:
\[ P = 2L + 2\left(\frac{121}{L}\right) = 2L + \frac{242}{L} \]
To find the minimum fencing needed, we ideally want \(L\) and \(W\) to be as close to each other as possible (for a rectangle, this will minimize perimeter for a given area). The dimensions that achieve this for an area of 121 ft² could be approximated by taking the square root:
\[ L \approx W \approx \sqrt{121} = 11 , \text{ft} \]
So if both sides were approximately 11 ft, the perimeter would be:
\[ P = 2(11) + 2(11) = 44 , \text{ft} \]
Thus, Kai would need approximately 44 feet of fencing to enclose his garden.
However, since the garden could have multiple configurations (e.g., \(L = 1\) and \(W = 121\)), the exact amount of fencing required can vary. In summary, for standard rectangular dimensions around the given area, Kai will need approximately 44 feet of fencing.