To find the length of the spring when no weights are attached, we can use the information given about the spring under different weights.
Let \( L \) be the length of the spring and \( W \) be the weight attached. We can use the two points given:
- When \( W = 2 \) pounds, \( L = 12 \) inches.
- When \( W = 5 \) pounds, \( L = 18 \) inches.
We can find the rate of change (slope) of the length of the spring with respect to the weight by first determining the change in length and the change in weight.
The change in length (from 12 inches to 18 inches) is:
\[ \Delta L = 18 - 12 = 6 \text{ inches} \]
The change in weight (from 2 pounds to 5 pounds) is:
\[ \Delta W = 5 - 2 = 3 \text{ pounds} \]
Now, we can find the rate of change (slope) of the length with respect to the weight:
\[ \text{Rate of change} = \frac{\Delta L}{\Delta W} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound} \]
This tells us that for every pound added to the weight, the length of the spring increases by 2 inches.
Now, we can express the relationship between the length of the spring and the weight attached:
Starting with the weight of 2 pounds, where the length is 12 inches, we can write the linear equation as follows. If \( L \) is the length and \( W \) is the weight, we can express \( L \) as a linear function of \( W \):
\[ L = mW + b \]
Where \( m \) is the slope (2 inches per pound) and \( b \) is the length of the spring when no weight is attached (the y-intercept).
Using the known point (2, 12):
\[ 12 = 2(2) + b \ 12 = 4 + b \ b = 12 - 4 \ b = 8 \]
Thus, the equation of the spring's length in terms of weight is:
\[ L = 2W + 8 \]
To find the length of the spring when no weight is attached (i.e., when \( W = 0 \)):
\[ L = 2(0) + 8 = 8 \text{ inches} \]
Therefore, the length of the spring when no weights are attached is 8 inches.