This question aims to find the **highest** and **lowest frequency** of the string of an **out-of-tune piano**. The hammer of this piano hits two strings and produces a beat of **7 Hz** and one of the strings of this piano is tuned to **131 Hz.**

The number of waves passing through a fixed point in a fixed amount of time is called **frequency**. Frequency is the **reciprocal of time.** For example, if a wave takes 3 seconds to pass through a fixed point, then it means its frequency is 3 Hz. The unit of frequency is **Hertz** and it is represented as **Hz**.

The high peaks in the graph show high frequency and the small peaks show low frequency. When an object experiences **periodic motion**, it creates vibrations of certain frequencies.

The **sound of a hammer** hitting the strings of the piano tells whether a piano is perfect or out of tune. When the hammer of a piano hits more than one string tunes to the same pitch then the piano is perfectly fine.

If the piano sounds **off-tune,** then the strings of the piano are out of tune. The sound of the piano tells a lot about the piano. Irritating loud sounds tell us that the piano needs tuning.

## Expert Answer

The given values of frequencies are:

\[ Frequency 1 = 7 Hz \]

\[ F_1 = 7 Hz \]

\[ Frequency 2 = 131 Hz \]

\[ F_2 = 131 Hz \]

The string has the highest frequency as follows:

\[ Highest Frequency = F_1 + F_2 \]

\[ Highest Frequency = 131 + 7 \]

\[ Highest Frequency = 138 Hz \]

The string has the lowest frequency as follows:

\[ Lowest Frequency = F_2 – F_1 \]

\[ Lowest Frequency = 131 – 7 \]

\[ Lowest Frequency = 124 Hz \]

## Numerical Results

The highest frequency of the string is $ 131 Hz $ and the lowest frequency of the string is $ 124 $ Hz.

## Example

If the frequencies of the two strings of a guitar are 16 Hz and 23 Hz, then find the highest and lowest frequencies.

The given values of frequencies are:

\[ Frequency 1 = 16 Hz \]

\[ F_1 = 16 Hz \]

\[ Frequency 2 = 23 Hz \]

\[ F_2 = 23 Hz \]

The string has the highest frequency as follows:

\[ Highest Frequency = F_1 + F_2 \]

\[ Highest Frequency = 16 + 23 Hz \]

\[ Highest Frequency = 39 Hz \]

The string has the lowest frequency as follows:

\[ Lowest Frequency = F_2 – F_1 \]

\[ Lowest Frequency = 23 – 16 Hz \]

\[ Lowest Frequency = 7 Hz \]

The highest frequency of the string is $ 39 Hz $ and the lowest frequency of the string is $ 7 Hz $.

If these strings produce the frequency of **12 Hz** and **10 Hz,** then the highest frequency is $22 Hz$.

*Image/Mathematical drawings are created in Geogebra*