on Mindy's guitar, the frequency of a vibrating string varies inversely as the length of the string. When a guitar string of "s" inches in length that vibrates at a frequency of 80 cycles per second is shortened to 6 inches in length, it vibrates at a frequency of 120 cycles per second. What is the value of "s"?

F1/F2 = L2/L1

80/120 = 6/s
80s = 720
s = 720/80 = 9

check:

is 80/120 = 6/9 ?
80/120 = 2/3
and
6/9 = 2/3

looking good!

To find the value of "s," we need to use the concept of inverse variation between frequency and length of the string. Inverse variation states that if one quantity increases, the other quantity decreases in proportion.

Let's start by setting up the equation for inverse variation:

Frequency = constant / Length

We can represent the constant of proportionality as "k":

Frequency = k / Length

Now, we can use the given information to form two equations:

Equation 1: Frequency1 = 80 cycles per second, Length1 = s inches
Equation 2: Frequency2 = 120 cycles per second, Length2 = 6 inches

Plugging these values into the equation, we get:

Equation 1: 80 = k / s
Equation 2: 120 = k / 6

To solve for "k," we can use Equation 2:

120 = k / 6
Multiply both sides by 6 to isolate "k":
k = 120 * 6
k = 720

Now that we have the value of "k," we can substitute it into Equation 1:

80 = 720 / s

To isolate "s," we can cross-multiply:

80s = 720
Divide both sides by 80 to solve for "s":
s = 720 / 80
s = 9

Therefore, the value of "s" is 9 inches.