#### Answer

The maximum value for each sound is 2 and hence the sound produced by each button has the same loudness.

#### Work Step by Step

The sound produced by the touch-tone pad is described by:
$y=\sin 2\pi lt+\sin 2\pi ht$
Where, $h$ is the high-frequency value and $l$ is the low-frequency value in cycles per second.
When touching button number 1,
$\begin{align}
& l=697\text{ cycles per second} \\
& h=1209\text{ cycles per second}
\end{align}$
So, the sound produced by 1 is described by:
$y=\sin 2\pi \left( 697 \right)t+\sin 2\pi \left( 1209 \right)t$
The maximum value of $y$ for the sound produced by touching 1 is:
${{y}_{\max \left( 1 \right)}}=2$
When touching the button number 2,
$\begin{align}
& l=697\text{ cycles per second} \\
& h=1336\text{ cycles per second}
\end{align}$
So, the sound produced by 2 is described by:
$y=\sin 2\pi \left( 697 \right)t+\sin 2\pi \left( 1336 \right)t$
The maximum value of $y$ for the sound produced by touching 2 is:
${{y}_{\max \left( 2 \right)}}=2$
When touching the button number 3,
$\begin{align}
& l=697\text{ cycles per second} \\
& h=1209\text{ cycles per second}
\end{align}$
So the sound produced by 3 is described by:
$y=\sin 2\pi \left( 697 \right)t+\sin 2\pi \left( 1477 \right)t$
The maximum value of $y$ for the sound produced by touching 3 is:
${{y}_{\max \left( 3 \right)}}=2$
When touching the button number 4,
$\begin{align}
& l=770\text{ cycles per second} \\
& h=1209\text{ cycles per second}
\end{align}$
So the sound produced by 4 is described by:
$y=\sin 2\pi \left( 770 \right)t+\sin 2\pi \left( 1209 \right)t$
The maximum value of $y$ for the sound produced by touching 4 is:
${{y}_{\max \left( 4 \right)}}=2$
When touching the button number 5,
$\begin{align}
& l=770\text{ cycles per second} \\
& h=1336\text{ cycles per second}
\end{align}$
So the produced by 5 is described by:
$y=\sin 2\pi \left( 770 \right)t+\sin 2\pi \left( 1336 \right)t$
The maximum value of $y$ for the sound produced by touching 5 is:
${{y}_{\max \left( 5 \right)}}=2$
When touching the button number 6,
$\begin{align}
& l=770\text{ cycles per second} \\
& h=1477\text{ cycles per second}
\end{align}$
So the sound produced by 6 is described by:
$y=\sin 2\pi \left( 770 \right)t+\sin 2\pi \left( 1477 \right)t$
The maximum value of $y$ for sound produced by touching 6 is:
${{y}_{\max \left( 6 \right)}}=2$
When touching button number 7,
$\begin{align}
& l=852\text{ cycles per second} \\
& h=1209\text{ cycles per second}
\end{align}$
So, the sound produced by 7 is described by:
$y=\sin 2\pi \left( 852 \right)t+\sin 2\pi \left( 1209 \right)t$
The maximum value of $y$ for the sound produced by touching 7 is:
${{y}_{\max \left( 7 \right)}}=2$
When touching button number 8,
$\begin{align}
& l=852\text{ cycles per second} \\
& h=1336\text{ cycles per second}
\end{align}$
So, the sound produced by 8 is described by:
$y=\sin 2\pi \left( 852 \right)t+\sin 2\pi \left( 1336 \right)t$
The maximum value of $y$ for the sound produced by touching 8 is:
${{y}_{\max \left( 8 \right)}}=2$
When touching button number 9,
$\begin{align}
& l=852\text{ cycles per second} \\
& h=1477\text{ cycles per second}
\end{align}$
So, the sound produced by 9 is described by:
$y=\sin 2\pi \left( 852 \right)t+\sin 2\pi \left( 1477 \right)t$
The maximum value of $y$ for the sound produced by touching 9 is:
${{y}_{\max \left( 9 \right)}}=2$
When touching button number 0,
$\begin{align}
& l=941\text{ cycles per second} \\
& h=1336\text{ cycles per second}
\end{align}$
So, the sound produced by 0 is described by:
$y=\sin 2\pi \left( 941 \right)t+\sin 2\pi \left( 1336 \right)t$
The maximum value of $y$ for the sound produced by touching 0 is:
${{y}_{\max \left( 0 \right)}}=2$
When touching button number *,
$\begin{align}
& l=941\text{ cycles per second} \\
& h=1209\text{ cycles per second}
\end{align}$
So, the sound produced by touching $*$ is described by:
$y=\sin 2\pi \left( 941 \right)t+\sin 2\pi \left( 1209 \right)t$
The maximum value of $y$ for the sound produced by touching $*$ is:
${{y}_{\max \left( * \right)}}=2$
When touching button number #,
$\begin{align}
& l=941\text{ cycles per second} \\
& h=1477\text{ cycles per second}
\end{align}$
So, the sound produced by touching $\#$ is described by:
$y=\sin 2\pi \left( 941 \right)t+\sin 2\pi \left( 1477 \right)t$
The maximum value of $y$ for the sound produced by touching $\#$ is:
${{y}_{\max \left( \# \right)}}=2$