First, we need to calculate the rate of change in length of the spring with respect to the weight.
Let's define:
- L1 = length of the spring when a 2-pound weight is attached (12 inches)
- L2 = length of the spring when a 5-pound weight is attached (18 inches)
- W1 = weight of 2 pounds
- W2 = weight of 5 pounds
The rate of change in length of the spring is given by the formula:
Rate of change = (L2 - L1) / (W2 - W1)
Substitute the given values:
Rate of change = (18 - 12) / (5 - 2)
Rate of change = 6 / 3
Rate of change = 2 inches/pound
Now, to find the length of the spring when no weights are attached:
Let L0 be the length of the spring with no weights attached.
We know:
- W0 = 0 pounds
The rate of change formula:
Rate of change = (L0 - L1) / (W0 - W1)
Substitute the rate of change value we found earlier:
2 = (L0 - 12) / (0 - 2)
2 = (L0 - 12) / (-2)
Now, cross multiply to solve for L0:
2 * -2 = L0 - 12
-4 = L0 - 12
L0 = -4 + 12
L0 = 8 inches
Therefore, the length of the spring when no weight is attached is 8 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. (1 point) inches
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