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.

Answer 1

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

Let x be the weight attached to the spring, and let y be the length of the spring. From the given information, we have two data points: (2, 12) and (5, 18). We can calculate the rate of change using these data points:

Rate of change = (change in y) / (change in x)
= (18 - 12) / (5 - 2)
= 6 / 3
= 2

This means that for every pound of weight added, the length of the spring increases by 2 inches.

Now, we can use this rate of change to find the length of the spring when no weights are attached. Let's let x = 0 be the weight of the spring with no weights attached:

y = 2x + b
12 = 2(2) + b
12 = 4 + b
b = 8

Therefore, the equation for the length of the spring when no weights are attached is y = 2x + 8.

Now, when x = 0, we have:

y = 2(0) + 8
y = 8

Therefore, the length of the spring when no weights are attached is 8 inches.