document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177187>Question 1177187</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Owen is jumping on a trampoline. When his feet hit the deck of the trampoline, the material depresses to a minimum height of 2cm. On average, Owen is reaching a maximum height of 200cm every 10 seconds. Determine the equation of a sinusoidal function that would model this situation, assuming Owen reaches his first maximum at 6 seconds. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850507 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.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=850507\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down how to model Owen's trampoline jumping with a sinusoidal function.\r<BR>\n" );
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document.write( "**Understanding the Information**\r<BR>\n" );
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document.write( "* **Minimum Height:** 2 cm (This is the vertical shift or midline of the function)<BR>\n" );
document.write( "* **Maximum Height:** 200 cm (This helps us find the amplitude)<BR>\n" );
document.write( "* **Period:** 10 seconds (Time for one complete jump cycle)<BR>\n" );
document.write( "* **First Maximum:** 6 seconds (This indicates a horizontal shift or phase shift)\r<BR>\n" );
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document.write( "**Building the Sinusoidal Function**\r<BR>\n" );
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document.write( "We'll use the general form of a sinusoidal function:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "h(t) = A * sin(B(t - C)) + D<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* **h(t):** Height of Owen at time t<BR>\n" );
document.write( "* **A:** Amplitude (half the difference between maximum and minimum height)<BR>\n" );
document.write( "* **B:** Determines the period (Period = 2π/B)<BR>\n" );
document.write( "* **C:** Horizontal shift (phase shift)<BR>\n" );
document.write( "* **D:** Vertical shift (midline)\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "1. **Amplitude (A):**<BR>\n" );
document.write( "   * A = (Maximum Height - Minimum Height) / 2<BR>\n" );
document.write( "   * A = (200 cm - 2 cm) / 2 = 99 cm\r<BR>\n" );
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document.write( "2. **Period (B):**<BR>\n" );
document.write( "   * Period = 10 seconds<BR>\n" );
document.write( "   * B = 2π / Period = 2π / 10 = π/5\r<BR>\n" );
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document.write( "3. **Horizontal Shift (C):**<BR>\n" );
document.write( "   * Owen reaches his first maximum at 6 seconds. Since the sine function starts at its midline and increases, we need to shift it to the right by 6 seconds.<BR>\n" );
document.write( "   * C = 6 seconds\r<BR>\n" );
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document.write( "4. **Vertical Shift (D):**<BR>\n" );
document.write( "   * D = Minimum Height + Amplitude<BR>\n" );
document.write( "   * D = 2 cm + 99 cm = 101 cm\r<BR>\n" );
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document.write( "**Equation of the Sinusoidal Function**\r<BR>\n" );
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document.write( "Putting it all together, the equation that models Owen's jumping is:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "h(t) = 99 * sin((π/5)(t - 6)) + 101<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Interpretation**\r<BR>\n" );
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document.write( "* This function models Owen's height (in cm) above the ground at any time t (in seconds).<BR>\n" );
document.write( "* The amplitude of 99 cm represents how high Owen jumps above the midline.<BR>\n" );
document.write( "* The period of 10 seconds represents the time it takes for one complete jump cycle.<BR>\n" );
document.write( "* The horizontal shift of 6 seconds represents the time it takes for Owen to reach his first maximum height.<BR>\n" );
document.write( "* The vertical shift of 101 cm represents the average height of Owen's jumps.\r<BR>\n" );
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document.write( "Let me know if you'd like to explore any variations of this scenario or have any other questions! </SPAN>\n" );
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