To find the length of the spring when no weights are attached, we can determine the rate of change in length with respect to the weight added.
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Calculate the change in length per pound of weight:
- The length of the spring goes from 12 inches (with a 2-pound weight) to 18 inches (with a 5-pound weight).
- The change in length: \(18 \text{ inches} - 12 \text{ inches} = 6 \text{ inches}\)
- The change in weight: \(5 \text{ pounds} - 2 \text{ pounds} = 3 \text{ pounds}\)
- Rate of change in length per pound: \[ \text{Rate} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound} \]
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Use the rate of change to determine the length of the spring with 0 pounds attached:
- When a 2-pound weight is applied, the spring stretches to 12 inches.
- The spring stretches an additional 2 inches for each additional pound. Since 2 pounds weight is applied, the additional stretch is: \[ 2 \text{ pounds} \times 2 \text{ inches per pound} = 4 \text{ inches} \]
- Therefore, the original (unstretched) length of the spring (when no weights are attached) can be found by subtracting this additional stretch from the length at 2 pounds: \[ \text{Original length} = 12 \text{ inches} - 4 \text{ inches} = 8 \text{ inches} \]
Thus, the length of the spring when no weights are attached is 8 inches.