document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178345>Question 1178345</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At a specific temperature, the pressure is 103.5 kPa at sea level and 89 kPa at h = 1,000 m. (Round your answers to one decimal place.)<BR>\n" );
document.write( "(a) What is the pressure at an altitude of 3500 m?<BR>\n" );
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document.write( "(b) What is the pressure at the top of a mountain that is 6251 m high? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850351 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850351\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Set up the Differential Equation**\r<BR>\n" );
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document.write( "The problem states that the rate of change of pressure (P) with respect to altitude (h) is proportional to P. This can be written as:\r<BR>\n" );
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document.write( "* dP/dh = kP\r<BR>\n" );
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document.write( "where k is the proportionality constant.\r<BR>\n" );
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document.write( "**2. Solve the Differential Equation**\r<BR>\n" );
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document.write( "This is a separable differential equation. We can solve it as follows:\r<BR>\n" );
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document.write( "* dP/P = k dh<BR>\n" );
document.write( "* ∫(dP/P) = ∫(k dh)<BR>\n" );
document.write( "* ln|P| = kh + C<BR>\n" );
document.write( "* P = Ae^(kh)\r<BR>\n" );
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document.write( "where A = e^C.\r<BR>\n" );
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document.write( "**3. Use the Given Information to Find A and k**\r<BR>\n" );
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document.write( "We are given:\r<BR>\n" );
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document.write( "* P = 103.5 kPa at h = 0 m (sea level)<BR>\n" );
document.write( "* P = 89 kPa at h = 1000 m\r<BR>\n" );
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document.write( "Using the first condition:\r<BR>\n" );
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document.write( "* 103.5 = Ae^(k * 0)<BR>\n" );
document.write( "* A = 103.5\r<BR>\n" );
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document.write( "So, the equation becomes:\r<BR>\n" );
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document.write( "* P = 103.5e^(kh)\r<BR>\n" );
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document.write( "Using the second condition:\r<BR>\n" );
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document.write( "* 89 = 103.5e^(1000k)<BR>\n" );
document.write( "* 89 / 103.5 = e^(1000k)<BR>\n" );
document.write( "* ln(89 / 103.5) = 1000k<BR>\n" );
document.write( "* k = ln(89 / 103.5) / 1000<BR>\n" );
document.write( "* k ≈ -0.0001484\r<BR>\n" );
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document.write( "Therefore, the equation is:\r<BR>\n" );
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document.write( "* P = 103.5e^(-0.0001484h)\r<BR>\n" );
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document.write( "**4. Calculate the Pressure at 3500 m**\r<BR>\n" );
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document.write( "* P = 103.5e^(-0.0001484 * 3500)<BR>\n" );
document.write( "* P ≈ 103.5e^(-0.5194)<BR>\n" );
document.write( "* P ≈ 103.5 * 0.5947<BR>\n" );
document.write( "* P ≈ 61.55 kPa\r<BR>\n" );
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document.write( "Rounded to one decimal place:\r<BR>\n" );
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document.write( "* P ≈ 61.6 kPa\r<BR>\n" );
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document.write( "**5. Calculate the Pressure at 6251 m**\r<BR>\n" );
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document.write( "* P = 103.5e^(-0.0001484 * 6251)<BR>\n" );
document.write( "* P ≈ 103.5e^(-0.9276)<BR>\n" );
document.write( "* P ≈ 103.5 * 0.3954<BR>\n" );
document.write( "* P ≈ 40.92 kPa\r<BR>\n" );
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document.write( "Rounded to one decimal place:\r<BR>\n" );
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document.write( "* P ≈ 40.9 kPa\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "(a) The pressure at an altitude of 3500 m is approximately **61.6 kPa**.\r<BR>\n" );
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document.write( "(b) The pressure at the top of a mountain that is 6251 m high is approximately **40.9 kPa**.<BR>\n" );
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