To find the length of the spring when no weights are attached, we can use the concept of rate of change.
Let x be the weight attached to the spring, and let y be the length of the spring. From the given information, we have two data points: (2, 12) and (5, 18). We can calculate the rate of change using these data points:
Rate of change = (change in y) / (change in x)
= (18 - 12) / (5 - 2)
= 6 / 3
= 2
This means that for every pound of weight added, the length of the spring increases by 2 inches.
Now, we can use this rate of change to find the length of the spring when no weights are attached. Let's let x = 0 be the weight of the spring with no weights attached:
y = 2x + b
12 = 2(2) + b
12 = 4 + b
b = 8
Therefore, the equation for the length of the spring when no weights are attached is y = 2x + 8.
Now, when x = 0, we have:
y = 2(0) + 8
y = 8
Therefore, the length of the spring when no weights are attached is 8 inches.
An illustration shows three coiled wire springs stretched to varying lengths. The first spring does not have any weights pulling it downward, and the length of the spring is marked as a question mark inches. The second spring has a 2-pound weight attached to the bottom, and its length is marked as 12 inches. The third spring has a 5-pound weight attached to the bottom and its length is marked as 18 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 rate of change to find the length of the spring when no weights are attached.
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