document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202197>Question 1202197</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Out at sea a fishing boat rises and falls due to the stormy waves.  The model h(t) = sin (36t) represents the displacement of the fishing boar, h(t), in metres at t seconds.<BR>\n" );
document.write( "a) Calculate the period.<BR>\n" );
document.write( "b) Determine the displacement at 30 s<BR>\n" );
document.write( "c) Determine the displacement at 12 s<BR>\n" );
document.write( "d) At what time, to the nearest tenth of a second, does the displacement first reach 0.7 m. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836853 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=836853\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I'll use x in place of t, and y in place of h(t).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The given equation is y = sin(36x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Compare it to the template y = A*sin(B(x-C))+D\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The variables are: <BR>\n" );
document.write( "|A| = amplitude<BR>\n" );
document.write( "B = helps determine the period, more on that later<BR>\n" );
document.write( "C = phase shift, which is the side to side shift<BR>\n" );
document.write( "D = up and down shift, useful to determine midline\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In this case,<BR>\n" );
document.write( "A = 1<BR>\n" );
document.write( "B = 36<BR>\n" );
document.write( "C = 0<BR>\n" );
document.write( "D = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When in radian mode, the period T is found through this formula<BR>\n" );
document.write( "T = 2pi/B<BR>\n" );
document.write( "T = 2pi/36<BR>\n" );
document.write( "T = pi/18<BR>\n" );
document.write( "Again, this only applies when working in radian mode.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you are in degree mode, then the period would be<BR>\n" );
document.write( "T = 360/B<BR>\n" );
document.write( "T = 360/36<BR>\n" );
document.write( "T = 10<BR>\n" );
document.write( "Since this result is much nicer, I have a feeling your teacher probably wants you to use degree mode. <BR>\n" );
document.write( "However, be sure to ask for clarification.<BR>\n" );
document.write( "The period represents how long each cycle takes. So the wave pattern repeats itself every 10 seconds.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red size=4>The period is 10 seconds, assuming you are in degree mode</font> (otherwise the period is pi/18 seconds in radian mode).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=============================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I've replaced t with x.<BR>\n" );
document.write( "I've replaced h(t) with y.<BR>\n" );
document.write( "x = time in seconds<BR>\n" );
document.write( "y = displacement, specifically how far up or down the boat is compared to the midline\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the rest of this homework, I'll assume your teacher wants degree mode.<BR>\n" );
document.write( "Plug x = 30 into the equation to determine y.<BR>\n" );
document.write( "y = sin(36x)<BR>\n" );
document.write( "y = sin(36*30)<BR>\n" );
document.write( "y = sin(1080)<BR>\n" );
document.write( "y = 0<BR>\n" );
document.write( "The displacement is 0.<BR>\n" );
document.write( "A displacement of 0 means the boat is at the midline.<BR>\n" );
document.write( "It is very likely the \"midline\" refers to the average sea level.<BR>\n" );
document.write( "Note that 1080 is a multiple of 360. It means angles 1080 degrees and 360 degrees are coterminal.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red size=4>0 meters</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=============================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (c)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Follow a similar set of steps we did for part (b).<BR>\n" );
document.write( "This time plug in x = 12.<BR>\n" );
document.write( "y = sin(36x)<BR>\n" );
document.write( "y = sin(36*12)<BR>\n" );
document.write( "y = sin(432)<BR>\n" );
document.write( "y = 0.9510565<BR>\n" );
document.write( "This represents how high the boat is compared to the midline. <BR>\n" );
document.write( "If the displacement was negative, then the boat would be below the midline.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red size=4>Approximately 0.9510565 meters above the midline</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=============================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (d)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The previous two parts had us plug in an x value to find y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This time we'll start with a known y value to find x. <BR>\n" );
document.write( "There will be infinitely many solutions if we do not restrict the domain (because sine is not one-to-one). <BR>\n" );
document.write( "However, we'll be looking for the smallest positive x solution, which represents the first occurrence.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll need the inverse sine, aka arcsine, to isolate x.<BR>\n" );
document.write( "Your calculator likely shows this as a button labeled <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin^{-1}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = sin(36x)<BR>\n" );
document.write( "0.7 = sin(36x)<BR>\n" );
document.write( "sin(36x) = 0.7<BR>\n" );
document.write( "36x = arcsin(0.7)+360*n or 36x = 180-arcsin(0.7)+360*n<BR>\n" );
document.write( "36x = 44.427004+360*n or 36x = 180-44.427004+360*n<BR>\n" );
document.write( "36x = 44.427004+360*n or 36x = 135.572996+360*n<BR>\n" );
document.write( "x = (44.427004+360*n)/36 or x = (135.572996+360*n)/36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "That represents the set of all possible solutions for x.<BR>\n" );
document.write( "n is any integer.<BR>\n" );
document.write( "The decimal values are approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let,<BR>\n" );
document.write( "A = (44.427004+360*n)/36<BR>\n" );
document.write( "B = (135.572996+360*n)/36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's a table of various integer n values, along with A and B as well.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >n</TD><TD >A</TD><TD >B</TD></tr><tr><TD >-2</TD><TD >-18.8</TD><TD >-16.2</TD></tr><tr><TD >-1</TD><TD >-8.8</TD><TD >-6.2</TD></tr><tr><TD >0</TD><TD ><font color=red size=4>1.2</font></TD><TD >3.8</TD></tr><tr><TD >1</TD><TD >11.2</TD><TD >13.8</TD></tr><tr><TD >2</TD><TD >21.2</TD><TD >23.8</TD></tr><tr><TD >3</TD><TD >31.2</TD><TD >33.8</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Each decimal value is approximate.<BR>\n" );
document.write( "Highlighted in red is the solution we're after.<BR>\n" );
document.write( "This is the smallest positive x solution that makes sin(36x) = 0.7 true when in degree mode.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I recommend using a graphing calculator such as GeoGebra or Desmos to verify the answer is correct.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here is a link to the interactive Desmos graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/oaieombfdm\">https://www.desmos.com/calculator/oaieombfdm</a><BR>\n" );
document.write( "The sine wave crosses the horizontal line at approximately (1.2, 0.7) to visually confirm x = 1.2 is the smallest positive x solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red size=4>1.2 seconds</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );