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.
Let's assume that the length of the spring, when no weights are attached, is L inches.
We are given that the length of the spring is 12 inches when a 2-pound weight is attached. This gives us the first data point: (2, 12).
We are also given that the length of the spring is 18 inches when a 5-pound weight is attached. This gives us the second data point: (5, 18).
We can calculate the rate of change, which is the change in length divided by the change in weight. This can be calculated using the formula:
Rate of change = (change in length) / (change in weight)
Using the two data points, we can calculate the rate of change as follows:
Rate of change = (18 - 12) / (5 - 2)
= 6 / 3
= 2 inches/pound
This means that for every pound of weight added, the length of the spring increases by 2 inches.
To find the length of the spring when no weights are attached (L), we can use the rate of change formula:
Rate of change = (change in length) / (change in weight)
2 inches/pound = (L - 12) / (0 - 2)
Simplifying the equation, we have:
2 = (L - 12) / (-2)
Multiplying both sides by -2, we get:
-4 = L - 12
Adding 12 to both sides, we get:
L = 8
Therefore, the length of the spring when no weights are attached is 8 inches.