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.

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.

(?) 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.