To determine the length of the spring when no weights are attached, we can analyze the change in length per pound of weight added, using the data provided.
- When a 2-pound weight is attached, the spring's length is 12 inches.
- When a 5-pound weight is attached, the spring's length is 18 inches.
First, we can find the change in length of the spring as the weight increases from 2 pounds to 5 pounds:
- Change in weight = 5 pounds - 2 pounds = 3 pounds
- Change in length = 18 inches - 12 inches = 6 inches
Now, we can find the rate of change of length per pound of weight:
\[ \text{Rate of change} = \frac{\text{Change in length}}{\text{Change in weight}} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound} \]
Next, we can use the information about the 2-pound weight to set up an equation for the lengths. Let \( L \) be the length of the spring when no weights are attached.
The length of the spring when a weight \( w \) is attached can be represented as:
\[ L + 2 \times 2 = 12 \text{ (for 2 pounds)} \] \[ L + 5 \times 2 = 18 \text{ (for 5 pounds)} \]
These can be simplified as follows:
- \( L + 4 = 12 \) → \( L = 12 - 4 = 8 \)
- \( L + 10 = 18 \) → \( L = 18 - 10 = 8 \)
In both cases, we find that the length of the spring when no weights are attached is 8 inches.
Thus, the answer is:
8 inches
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