Question

Use the image to answer the question.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

Answer 1

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

We have two points:
1. When 2 pounds is attached, the length is 12 inches: (2, 12)
2. When 5 pounds is attached, the length is 18 inches: (5, 18)

We can find the rate of change (slope) of the length with respect to the weight:

1. Calculate the change in length and the change in weight:
\[
\text{Change in length} = 18 - 12 = 6 \text{ inches}
\]
\[
\text{Change in weight} = 5 - 2 = 3 \text{ pounds}
\]

2. Now, find the rate of change (slope):
\[
\text{Rate of change} = \frac{\text{Change in length}}{\text{Change in weight}} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound}
\]

3. Next, we can use the slope to find the y-intercept (the length of the spring when no weight is attached). We can use one of the points to write the equation of the line in the slope-intercept form \( y = mx + b \):
\[
L = 2w + b
\]
where \( L \) is the length, \( w \) is the weight, and \( b \) is the length when no weight is attached.

4. Using the point (2, 12):
\[
12 = 2(2) + b
\]
\[
12 = 4 + b
\]
\[
b = 12 - 4 = 8
\]

So, when there are no weights attached (0 pounds), the length of the spring is:
\[
L = 2(0) + 8 = 8 \text{ inches}
\]

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