Hooke's Law states that the distance a spring will stretch beyond its natural length varies directly with the force applied to the spring. A force of 12 pounds is needed to stretch a certain spring 9 inches beyond its natural length.

Find a formula that models the length a spring will stretch beyond its natural length in inches as a function of the force applied to the string in pounds. (All I could figure out is that the input is the weight and the output is length. I don't know how to find the formula though, please help!!)

Just translate the English into Math ...

" the distance a spring will stretch ...varies directly with the force applied to the spring"
---> d = k(1/F) , where d is the distance, and F is the force
use the given: d= 9, F = 12

9 = k(1/12)
k = 108

thus d = 108/F

now wasn't that easy ?

To find the formula that models the length a spring will stretch beyond its natural length as a function of the force applied to the spring, we can use Hooke's Law. Hooke's Law states that the distance a spring stretches or compresses is directly proportional to the force applied to it.

Let's analyze the given information: A force of 12 pounds is needed to stretch the spring 9 inches beyond its natural length. This means that when the force applied is 12 pounds, the length stretched is 9 inches.

We can use this relationship to find the proportionality constant, which will help us establish the formula. Let's denote the force as F and the length stretched as x.

Using the given data, we have the following equation:

12 pounds = k * 9 inches

Here, k represents the proportionality constant.

To solve for k, divide both sides of the equation by 9 inches:

12 pounds / 9 inches = k

Now, we have the value of k, which is the constant of proportionality. We find:

k ≈ 1.333 pounds/inch

The formula that models the length a spring will stretch beyond its natural length in inches as a function of the force applied to the spring in pounds can now be written as:

x = k * F

Substituting the value of k, the formula becomes:

x ≈ 1.333 * F

Therefore, the formula is:

Length stretched beyond natural length (in inches) ≈ 1.333 * Force applied to the spring (in pounds)