Let's denote the length of the spring with no weight attached as $x$ inches.
From the given information, we can form two equations:
When a 2-pound weight is attached, the length of the spring is 12 inches:
$x + 2 = 12$
When a 5-pound weight is attached, the length of the spring is 18 inches:
$x + 5 = 18$
Solving these equations, we can find the value of $x$:
$x = 12 - 2 = 10$ (from the first equation)
$x = 18 - 5 = 13$ (from the second equation)
Since the rate of change of the spring length is linear, we can take the average of the two values of $x$ to find the length of the spring when no weight is attached:
$\frac{10 + 13}{2} = \frac{23}{2} = 11.5$
Therefore, the length of the spring when no weight is attached is 11.5 inches.
A spring has a length of 12 inches when a 2-pound weight is attached, and a length of 18 inches when a 5-pound weight is attached. Use the rate of change to find the length of the spring when no weight were attached.
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