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 answer

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.