Question 414299
direct and inverse proportion formulas are shown below:

y = k*x (direct proportion)

y = k/x (inverse proportion)

Since you are dealing with inverse proportion, then the second formula applies.

k is the constant of proportionality.

you are given that a string of 48 cm vibrates 192 cycles per second.

you can use this fact to find the constant of proportionality.

let y = number of cycles per second.
let x = length of the string.

y = k/x becomes 192 = k/48

multiply both sides of this equation by 48 to get:

k = 192 * 48 = 9216

you apply that constant of proportionality when x = 32 cm.

you get y = 9216 / 32 = 288

the string will vibrate 288 cycles per second when the length of the string is 32 cm.

you can see that the proportion is inverse because, as the string length increases, the number of vibrations per second decreases.

you can also see that the ratio of the short string to the long string is inversely proportional as shown in the following table.

<pre>

long string          short string        ratio of short string to long string

l = 48               l = 32              32/48 = 2/3

vbs = 192            vbs = 288           288/192 = 3/2

</pre>

l = length
vbs = vibrations per second