document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=868347>Question 868347</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During an experiment, the intensity i(t) of the electric current of a device as a function of time t elapsed since the beginning of the experiment is given by:<BR>\n" );
document.write( "i(t)=6sin((pi*t/12)+(2*pi/3))+6<BR>\n" );
document.write( "The device emits a sound signal each time the current’s intensity is equal to 9.<BR>\n" );
document.write( "The experiment lasts 120 seconds.<BR>\n" );
document.write( "How many sound signals does the device emit during the experiment?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #523834 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=523834\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The device emits a signal every time the intensity, i(t), is equal to 9. So we need to find out how many times it equals 9. This is within the interval t = 0 to t = 120 since the experiment lasts 120 seconds.\r<BR>\n" );
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document.write( "So that means t is in the interval [0,120]. This is important to selecting the right solutions.\r<BR>\n" );
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document.write( "Plug in i(t) = 9 and solve for t\r<BR>\n" );
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document.write( "i(t)=6sin((pi*t/12)+(2*pi/3))+6\r<BR>\n" );
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document.write( "9=6sin((pi*t/12)+(2*pi/3))+6\r<BR>\n" );
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document.write( "9-6=6sin((pi*t/12)+(2*pi/3))\r<BR>\n" );
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document.write( "3=6sin((pi*t/12)+(2*pi/3))\r<BR>\n" );
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document.write( "6sin((pi*t/12)+(2*pi/3)) = 3\r<BR>\n" );
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document.write( "sin((pi*t/12)+(2*pi/3)) = 3/6\r<BR>\n" );
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document.write( "sin((pi*t/12)+(2*pi/3)) = 1/2\r<BR>\n" );
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document.write( "(pi*t/12)+(2*pi/3) = arcsin(1/2) or (pi*t/12)+(2*pi/3) = pi - arcsin(1/2)\r<BR>\n" );
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document.write( "(pi*t/12)+(2*pi/3) = pi/6 or (pi*t/12)+(2*pi/3) = pi - pi/6\r<BR>\n" );
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document.write( "(pi*t/12)+(2*pi/3) = pi/6 or (pi*t/12)+(2*pi/3) = 5pi/6\r<BR>\n" );
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document.write( "(pi*t/12)+(2*pi/3) = pi/6 + 2pi*n or (pi*t/12)+(2*pi/3) = 5pi/6 + 2pi*n\r<BR>\n" );
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document.write( "pi*t/12 = pi/6 + 2pi*n - 2*pi/3 or pi*t/12 = 5pi/6 + 2pi*n - 2*pi/3\r<BR>\n" );
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document.write( "t = 12/pi(pi/6 + 2pi*n - 2*pi/3) or t = 12/pi(5pi/6 + 2pi*n - 2*pi/3)\r<BR>\n" );
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document.write( "t = 2 + 24n - 8 or t = 10 + 24n - 8\r<BR>\n" );
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document.write( "t = 24n - 6 or t = 24n + 2\r<BR>\n" );
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document.write( "So from all of that work above, we have determined that the solutions to i(t) = 9 are t = 24n - 6 or t = 24n + 2 where n is an integer.\r<BR>\n" );
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document.write( "Now let's plug in integral values for n into the two equations.\r<BR>\n" );
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document.write( "Let's pick on the first equation t = 24n - 6\r<BR>\n" );
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document.write( "If n = 0, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(0) - 6<BR>\n" );
document.write( "t = -6 ... out of range of the interval [0,120]\r<BR>\n" );
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document.write( "If n = 1, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(1) - 6<BR>\n" );
document.write( "t = 18\r<BR>\n" );
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document.write( "If n = 2, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(2) - 6<BR>\n" );
document.write( "t = 42\r<BR>\n" );
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document.write( "If n = 3, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(3) - 6<BR>\n" );
document.write( "t = 66\r<BR>\n" );
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document.write( "If n = 4, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(4) - 6<BR>\n" );
document.write( "t = 90\r<BR>\n" );
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document.write( "If n = 5, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(5) - 6<BR>\n" );
document.write( "t = 114\r<BR>\n" );
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document.write( "If n = 6, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(6) - 6<BR>\n" );
document.write( "t = 138 ... out of range of the interval [0,120]\r<BR>\n" );
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document.write( "If n = 7, then t is...<BR>\n" );
document.write( "t = 24n - 6<BR>\n" );
document.write( "t = 24(7) - 6<BR>\n" );
document.write( "t = 162 ... out of range of the interval [0,120]\r<BR>\n" );
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document.write( "So we see that as n increases, we're getting larger and larger t values that will be outside the interval [0,120]. So we can stop for this equation.\r<BR>\n" );
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document.write( "So far, the solutions are: 18, 42, 66, 90, 114\r<BR>\n" );
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document.write( "Repeat for the other equation\r<BR>\n" );
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document.write( "If n = -1, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(-1) + 2<BR>\n" );
document.write( "t = -22 ... out of range of the interval [0,120]\r<BR>\n" );
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document.write( "If n = 0, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(0) + 2<BR>\n" );
document.write( "t = 2\r<BR>\n" );
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document.write( "If n = 1, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(1) + 2<BR>\n" );
document.write( "t = 26\r<BR>\n" );
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document.write( "If n = 2, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(2) + 2<BR>\n" );
document.write( "t = 50\r<BR>\n" );
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document.write( "If n = 3, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(3) + 2<BR>\n" );
document.write( "t = 74\r<BR>\n" );
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document.write( "If n = 4, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(4) + 2<BR>\n" );
document.write( "t = 98\r<BR>\n" );
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document.write( "If n = 5, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(5) + 2<BR>\n" );
document.write( "t = 122 ... out of range of the interval [0,120]\r<BR>\n" );
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document.write( "If n = 6, then t is...<BR>\n" );
document.write( "t = 24n + 2<BR>\n" );
document.write( "t = 24(6) + 2<BR>\n" );
document.write( "t = 146 ... out of range of the interval [0,120]\r<BR>\n" );
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document.write( "The other set of solutions are: 2, 26, 50, 74, 98\r<BR>\n" );
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document.write( "Put them all together to get this final set of solutions:  18, 42, 66, 90, 114, 2, 26, 50, 74, 98\r<BR>\n" );
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document.write( "Here is that list sorted: 2, 18, 26, 42, 50, 66, 74, 90, 98, 114\r<BR>\n" );
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document.write( "There are 10 solutions listed above, so it emits a signal exactly 10 times. </SPAN>\n" );
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