To find the length of the spring when no weights are attached, we can use the concept of the rate of change (slope) based on the relationship between the weight and the length of the spring.
We know:
- A 2-pound weight results in a length of 12 inches.
- A 5-pound weight results in a length of 18 inches.
First, we can find the rate of change (slope) between the two points (2 pounds, 12 inches) and (5 pounds, 18 inches):
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Calculate the change in length (y): \[ \Delta y = 18 - 12 = 6 \text{ inches} \]
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Calculate the change in weight (x): \[ \Delta x = 5 - 2 = 3 \text{ pounds} \]
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Calculate the rate of change (slope): \[ \text{slope} = \frac{\Delta y}{\Delta x} = \frac{6}{3} = 2 \]
This means for every pound added, the spring stretches 2 inches.
Next, we can express the relationship between weight and length in the form \( L = mx + b \), where \( L \) is the length, \( x \) is the weight, \( m \) is the slope (rate of change), and \( b \) is the length when there is no weight attached.
Using one of the points, let's plug in the values. We can use the point (2, 12):
\[ 12 = 2(2) + b \] \[ 12 = 4 + b \] \[ b = 12 - 4 = 8 \]
Thus, when no weights are 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.