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.

(?) inches

Answer 1

To find the length of the spring when no weights are attached, we can use the information given about the lengths of the spring at different weights to set up a linear relationship.

1. **Identify the data points:**
- (2 pounds, 12 inches)
- (5 pounds, 18 inches)

2. **Calculate the slope (rate of change) of the line:**
The slope \( m \) is calculated using the formula:
\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \( (x_1, y_1) = (2, 12) \) and \( (x_2, y_2) = (5, 18) \).

\[
m = \frac{18 - 12}{5 - 2} = \frac{6}{3} = 2
\]

3. **Write the equation of the line:**
We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Using the point (2, 12) and the slope \( m = 2 \):
\[
y - 12 = 2(x - 2)
\]
Simplifying this, we get:
\[
y - 12 = 2x - 4
\]
\[
y = 2x + 8
\]

4. **Find the length when no weights are attached:**
To find the length of the spring when no weights are attached, substitute \( x = 0 \) (0 pounds):
\[
y = 2(0) + 8 = 8
\]

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