To find the length of the spring when no weights are attached, we can use the information given about the weights and the resulting lengths.
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We have two data points:
- For a 2-pound weight, the length is 12 inches.
- For a 5-pound weight, the length is 18 inches.
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We can find the rate of change (slope) of the length with respect to the weight applied: \[ \text{Rate of Change} = \frac{\text{Change in Length}}{\text{Change in Weight}} = \frac{18 - 12}{5 - 2} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound} \]
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Now that we have the rate of change, we can use one of the points to find the length when no weight (0 pounds) is applied. We can use the point with 2 pounds: \[ \text{Length} = \text{Initial Length} + \text{Rate of Change} \times \text{Weight} \] Rearranging gives us: \[ \text{Initial Length} = \text{Length} - \text{Rate of Change} \times \text{Weight} \]
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Substituting the values using the 2-pound weight: \[ \text{Initial Length} = 12 \text{ inches} - (2 \text{ inches/pound} \times 2 \text{ pounds}) \] \[ \text{Initial Length} = 12 \text{ inches} - 4 \text{ inches} = 8 \text{ inches} \]
Thus, the length of the spring when no weights are attached is 8 inches.