Question

There are 7 notes of "A" on a standard piano. When the third A is struck, the string associated with the key vibrates 220 times per second. The next A above the given A vibrates twice as fast. An exponential function with a base of two is used to determine the frequency of the 11 notes between the two A's.

Find this function. Hint: The function is of the form f(x) = k · 2(cx),where k and c are constants. Also f(0) = 220 and f(12) = 440.

Answer 1

The function is of the form $f(x) = k$ $·2^{cx}$, so we need to find $k$ and $c$

Also we know $f(0) = 220, f(12) = 440$

Create the system of equations:

$220 = k · 2^{c*0}$

$440 = k · 2^{c*12}$

From first equations: $k = 220$

From second equations:

$440 = 220 * 2^{12c}$

$2 = 2^{12c}$

$12c = 1$

$c=\frac{1}{12}$

So our function is: $f(x) = 220 * 2^{\frac{1}{12}x}$