Question

The frequency f in vibrations per second of the sound of a struck piano key can be expressed as the exponential equation f = 27.5 · 4(n - 1)/24, where n is the number of the key on the standard 88-key piano, with the leftmost key being key 1. What is the frequency (in vibrations per second) of middle c, the 40th key on the keyboard? (Round your answer to the nearest whole number.)

Answer 1

The frequency of middle C on a piano (the 40th key) is found by substituting the key number into the given exponential formula and performing the calculation, then rounding to the nearest whole number.

Step-by-step explanation:

The frequency f in vibrations per second of the sound of a struck piano key can be calculated with the given exponential equation f = 27.5 · 4(n - 1)/24, where n is the number of the key on a standard 88-key piano. To find the frequency of middle C, which is the 40th key on the keyboard, we substitute n with 40 and solve the equation:

f = 27.5 · 4(40 - 1)/24

After performing the arithmetic within the exponential part of the equation, we get:

f = 27.5 · 4(39)/24

We then simplify this to get the frequency. Rounded to the nearest whole number, the frequency of the 40th key, middle C, will be given as a specific value.