PIANO The frequency F(n) measured in vibrations per second...

Question

PIANO The frequency F(n), measured in vibrations per second, of a note on a piano is represented by the function F(n)=440(1.0595)n, where n is the number of notes above A-440. Each black or white key represents one note.

a. How many notes do you take to travel from A-440 to A?

b. What is the frequency of A?

c. Discribe the relationship between the number of notes between A-440 and A and the increase in frequency.

d. determine if the function F(n) is linear. Justify your answer.

 

the underlined n is supposed to be the exponent but I can't type those.

Answer 1

Assuming you are going to the A above A-440.  The next higher A has twice the frequency of A-440. So,

 

880=440(1.0595^n)

 

1.0595^n=880/440=2

 

Take the log of both sides

 

n log(1.0595) = log (2)    (log (1.0595^n)=n log(1.0595)

 

This will give n=12 (or very close to it)

 

If you go down to the lower A of frequency 220 you will get n=-12

 

So there are 12 notes from A-440 to the next A above or below.

 

The formula strictly should be F(n)=440(1.0595^n) if n is the no. of notes above A-440

 

Hope this much helps. 

Corrected on 8/30/18 as per comment received