Use the image to answer the question.
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.
The rate of change can be calculated by finding the change in length of the spring for a change in weight.
For the first spring (no weights attached) to the second spring (2-pound weight attached):
Change in weight = 2 pounds - 0 pounds = 2 pounds
Change in length = 12 inches - ? inches = 12 inches - ? inches = ? inches
Rate of change = Change in length / Change in weight = ? inches / 2 pounds = ? inches/pound
For the second spring (2-pound weight attached) to the third spring (5-pound weight attached):
Change in weight = 5 pounds - 2 pounds = 3 pounds
Change in length = 18 inches - 12 inches = 6 inches
Rate of change = Change in length / Change in weight = 6 inches / 3 pounds = 2 inches/pound
Now, using the rate of change calculated, we can find the length of the first spring with no weights attached:
Rate of change = 2 inches/pound
Change in weight = 0 pounds (no weights attached)
Change in length = ? inches
? inches = 0 pounds * 2 inches/pound = 0 inches
Therefore, the length of the spring with no weights attached is 0 inches.