To find the length of the spring when no weights are attached, we can use the concept of linear relationships.
We have two points:
- When 2 pounds is attached, the length is 12 inches: (2, 12)
- When 5 pounds is attached, the length is 18 inches: (5, 18)
We can find the rate of change (slope) of the length with respect to the weight:
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Calculate the change in length and the change in weight: \[ \text{Change in length} = 18 - 12 = 6 \text{ inches} \] \[ \text{Change in weight} = 5 - 2 = 3 \text{ pounds} \]
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Now, find the rate of change (slope): \[ \text{Rate of change} = \frac{\text{Change in length}}{\text{Change in weight}} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound} \]
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Next, we can use the slope to find the y-intercept (the length of the spring when no weight is attached). We can use one of the points to write the equation of the line in the slope-intercept form \( y = mx + b \): \[ L = 2w + b \] where \( L \) is the length, \( w \) is the weight, and \( b \) is the length when no weight is attached.
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Using the point (2, 12): \[ 12 = 2(2) + b \] \[ 12 = 4 + b \] \[ b = 12 - 4 = 8 \]
So, when there are no weights attached (0 pounds), the length of the spring is: \[ L = 2(0) + 8 = 8 \text{ inches} \]
Therefore, the length of the spring when no weights are attached is 8 inches.
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