To find the length of the spring when no weights are attached, we can establish a linear relationship based on the given data points.
We have:
- A 2-pound weight results in a length of 12 inches.
- A 5-pound weight results in a length of 18 inches.
First, we'll calculate the change in length (ΔL) and the change in weight (ΔW):
- From 2 pounds to 5 pounds, the weight increases by: \[ 5 - 2 = 3 \text{ pounds} \]
- From 12 inches to 18 inches, the length increases by: \[ 18 - 12 = 6 \text{ inches} \]
Now we can find the rate of change of length with respect to weight: \[ \text{Rate of change} = \frac{\Delta L}{\Delta W} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound} \]
Next, we can use this rate of change to express the length of the spring as a function of weight. We can use the point (2 pounds, 12 inches) to find the relationship. We can express the length \(L\) as: \[ L = k \cdot W + b \]
Where:
- \(k\) is the rate of change (2 inches per pound),
- \(W\) is the weight, and
- \(b\) is the length when no weight is applied.
Plugging in one of the points (using the 2-pound weight): \[ 12 = 2 \cdot 2 + b \] \[ 12 = 4 + b \] \[ b = 12 - 4 = 8 \text{ inches} \]
Thus, the length of the spring when no weights are attached (0 pounds) is: \[ \boxed{8} \text{ inches} \]