Question

a spring has a length of 12 inches with a 2 pound weight, and a length of 18 inches with a 5 pound weight, use rate of change to find the length of the spring with no weight

Answer 1

To find the length of the spring with no weight, we can use the linear relationship between the weight and the length of the spring. This relationship can be described using two points based on the information given:

1. When the weight is 2 pounds, the length of the spring is 12 inches. This gives us the point \((2, 12)\).
2. When the weight is 5 pounds, the length of the spring is 18 inches. This gives us the point \((5, 18)\).

Next, we can find the slope (rate of change) of the line that connects these two points. The slope \(m\) can be calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting in our values:

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

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

Next, we can use the slope to create a linear equation in the form:

\[
L(w) = mw + b
\]

where \(L(w)\) is the length of the spring in inches, \(w\) is the weight in pounds, \(m\) is the slope we calculated (2), and \(b\) is the y-intercept (the length of the spring when there is no weight).

To find \(b\), we can use one of the points. Let's use the point \((2, 12)\):

\[
12 = 2(2) + b
\]
\[
12 = 4 + b
\]
\[
b = 12 - 4 = 8
\]

So, the equation that describes the length of the spring with respect to the weight is:

\[
L(w) = 2w + 8
\]

Now, to find the length of the spring with no weight (when \(w = 0\)):

\[
L(0) = 2(0) + 8 = 8
\]

Thus, the length of the spring with no weight is **8 inches**.