We are told that the frequency of sound emitted by a string is inversely proportional to its length. Also, the frequency is 256Hz when the length is 50cm.
Note that when variables vary inversely, as one variable increases the other variable decreases and vice versa.
When two variables are inversely proportional (or vary inversely) the variables are related in the following manner: `y=k/x ` where the variables are x,y and k is called the constant of proportionality. Here we have `f=k/l ` where f is the frequency in hertz, l is the length in centimeters, and k is the constant of proportionality. Note that k will depend on outside factors such as the material of the string, the medium it is in, etc..., but k will be constant for the given situation.
We are given that when f=256, l=50, so we can solve for k:
`256=k/50 ==> k=50(256)=12800 `
(a) If l=40 we have `f=12800/40 ==> f=320 ` ; so when the length is 40cm the frequency is 320Hz.
This agrees with the knowledge that as the length gets shorter the frequency should increase.
(b) If f=400 then `400=12800/l ==> l=12800/400 ==> l=32 ` ; thus when the frequency is 400Hz, the length will be 32cm.