document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179653>Question 1179653</A>: <I><SPAN STYLE=\"word-break: break-all;\"> https://gyazo.com/848645a235363970dae8196a58a2bbdc </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850199 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850199\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down the problem step-by-step.\r<BR>\n" );
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document.write( "**1. Determine the Minimum Height of the Pendulum**\r<BR>\n" );
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document.write( "* The rope length is 3 meters.<BR>\n" );
document.write( "* The ceiling height is 4 meters.<BR>\n" );
document.write( "* The angle of the widest swing is π/3.<BR>\n" );
document.write( "* When the pendulum is at its lowest point (vertical), the height from the ceiling is 3 meters.<BR>\n" );
document.write( "* When the pendulum swings to its widest point, it forms a triangle with the vertical.<BR>\n" );
document.write( "* The vertical component of the rope at the widest swing is 3 * cos(π/3) = 3 * (1/2) = 1.5 meters.<BR>\n" );
document.write( "* Therefore, the change in vertical height from the lowest point to the widest point is 3 - 1.5 = 1.5 meters.<BR>\n" );
document.write( "* The lowest point of the pendulum is 4 - 3 = 1 meter from the ground.\r<BR>\n" );
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document.write( "**2. Determine the Amplitude**\r<BR>\n" );
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document.write( "* The amplitude is the change in height from the lowest point to the widest point, which is 1.5 meters.\r<BR>\n" );
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document.write( "**3. Determine the Vertical Shift**\r<BR>\n" );
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document.write( "* The vertical shift is the height of the pendulum when it is at its lowest point, which is 1 meter from the ground.<BR>\n" );
document.write( "* Since the lowest point is 1 meter, the midline of the cosine wave will be 1 + 1.5 = 2.5 meters.\r<BR>\n" );
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document.write( "**4. Determine the Period and Angular Frequency**\r<BR>\n" );
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document.write( "* The pendulum swings out to its widest position in 2 seconds. This is half of the period.<BR>\n" );
document.write( "* Therefore, the full period is 4 seconds.<BR>\n" );
document.write( "* The angular frequency (ω) is 2π / period = 2π / 4 = π/2.\r<BR>\n" );
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document.write( "**5. Determine the Phase Shift**\r<BR>\n" );
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document.write( "* The pendulum is at its lowest point when t = 0.<BR>\n" );
document.write( "* A cosine function starts at its maximum value. To model the lowest point at t=0, we need to use a negative cosine.<BR>\n" );
document.write( "* There is no horizontal phase shift.\r<BR>\n" );
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document.write( "**6. Model the Height Function**\r<BR>\n" );
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document.write( "* The general form of a cosine function is: y = A * cos(ωt + φ) + D<BR>\n" );
document.write( "* A = Amplitude = 1.5<BR>\n" );
document.write( "* ω = Angular frequency = π/2<BR>\n" );
document.write( "* φ = Phase shift = 0<BR>\n" );
document.write( "* D = Vertical shift = 2.5<BR>\n" );
document.write( "* Since we need a negative cosine, we have:<BR>\n" );
document.write( "* h(t) = -1.5 * cos(π/2 * t) + 2.5\r<BR>\n" );
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document.write( "**Final Answer**\r<BR>\n" );
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document.write( "The height of the pendulum from the ground as a function of time is:\r<BR>\n" );
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document.write( "h(t) = -1.5cos(πt/2) + 2.5<BR>\n" );
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