## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 5 - Section 5.4 - Product-to-Sum and Sum-to-Product Formulas - Exercise Set - Page 690: 44

#### 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: ${{y}_{\max \left( \# \right)}}=2$

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