Question

Question

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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.

(1 point)
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Answer 1

To find the length of the spring when no weights are attached, we can use the rate of change formula:

Rate of change = (Change in y) / (Change in x)

In this case, our y values are the lengths of the spring and our x values are the weights attached.

For the first spring with no weight attached, the length is represented by a question mark inches. Let's call this length L1.

For the second spring with a 2-pound weight attached, the length is 12 inches. Let's call this length L2.

For the third spring with a 5-pound weight attached, the length is 18 inches. Let's call this length L3.

Using the rate of change formula:

Rate of change = (L3 - L2) / (5 - 2)

Rate of change = (18 - 12) / (5 - 2)

Rate of change = 6 / 3

Rate of change = 2

Now, to find the length of the spring when no weights are attached, we use the rate of change with one of the lengths:

Rate of change = (L1 - L2) / (0 - 2)

2 = (L1 - 12) / -2

2(-2) = -2(L1 - 12)

-4 = -2L1 + 24

-4 - 24 = -2L1

-28 = -2L1

L1 = 14

Therefore, the length of the spring when no weights are attached is 14 inches.