Final answer:
To find the length of the spring when no weights are attached, use the concept of rate of change and the given data points to find the equation of the line. Then, plug in x = 0 to find the length of the spring when no weights are attached.
Step-by-step explanation:
To find the length of the spring when no weights are attached, we can use the concept of rate of change. The rate of change of the spring's length with respect to the weight attached is a constant value known as the force constant of the spring. We can use this force constant to find the length of the spring when no weights are attached.
Let x be the weight attached in pounds, and let y be the length of the spring in inches. From the given information, we have the following data points: (2, 12) and (5, 18). We can use these points to find the equation of the line that represents the relationship between weight and length.
Using the equation of a line, y = mx + b, where m is the rate of change (force constant) and b is the y-intercept (unloaded length of the spring), we can solve for b. Plugging in the values (2, 12) and (5, 18), we can find the equation y = 2x + 8.5. To find the length of the spring when no weights are attached (x = 0), we plug in x = 0 into the equation and solve for y. The length of the spring when no weights are attached is 8.5 inches.