document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186380>Question 1186380</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 	Ridership on a small regional airline has been declining, approximately at a linear rate. In 1981 the number of passengers was 245,000; in 1986 the number was 215,000. If n equals the number of passengers using the airline per year and t equals time measured in year (t=0 for 1981)<BR>\n" );
document.write( "	Determine the linear estimating function n=f(t)<BR>\n" );
document.write( "	Interpret the meaning of the slope<BR>\n" );
document.write( "	What is the number of riders expected to equal in the year 2000?<BR>\n" );
document.write( "	It is estimated that the airline will go out of business if ridership falls below 180,000.  According to your function in part a, when will this happen?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849448 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=849448\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**1. Determine the linear estimating function n = f(t):**\r<BR>\n" );
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document.write( "We are given two points on the line:\r<BR>\n" );
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document.write( "*   (t₁, n₁) = (0, 245000)  (1981)<BR>\n" );
document.write( "*   (t₂, n₂) = (5, 215000)  (1986)\r<BR>\n" );
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document.write( "First, calculate the slope (m):\r<BR>\n" );
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document.write( "m = (n₂ - n₁) / (t₂ - t₁)<BR>\n" );
document.write( "m = (215000 - 245000) / (5 - 0)<BR>\n" );
document.write( "m = -30000 / 5<BR>\n" );
document.write( "m = -6000\r<BR>\n" );
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document.write( "Now, use the point-slope form of a linear equation with one of the points (e.g., (0, 245000)):\r<BR>\n" );
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document.write( "n - n₁ = m(t - t₁)<BR>\n" );
document.write( "n - 245000 = -6000(t - 0)<BR>\n" );
document.write( "n = -6000t + 245000\r<BR>\n" );
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document.write( "So, the linear estimating function is:  **n(t) = -6000t + 245000**\r<BR>\n" );
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document.write( "**2. Interpret the meaning of the slope:**\r<BR>\n" );
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document.write( "The slope, -6000, represents the average *decrease* in the number of passengers per year.  In other words, the airline's ridership is declining by approximately 6,000 passengers each year.\r<BR>\n" );
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document.write( "**3. What is the number of riders expected in the year 2000?**\r<BR>\n" );
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document.write( "The year 2000 is 19 years after 1981 (2000 - 1981 = 19), so t = 19.  Plug this into the equation:\r<BR>\n" );
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document.write( "n(19) = -6000 * 19 + 245000<BR>\n" );
document.write( "n(19) = -114000 + 245000<BR>\n" );
document.write( "n(19) = 131000\r<BR>\n" );
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document.write( "Therefore, the number of riders expected in the year 2000 is **131,000**.\r<BR>\n" );
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document.write( "**4. When will ridership fall below 180,000?**\r<BR>\n" );
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document.write( "We need to find the value of t when n(t) is less than 180,000:\r<BR>\n" );
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document.write( "180000 = -6000t + 245000<BR>\n" );
document.write( "-65000 = -6000t<BR>\n" );
document.write( "t = -65000 / -6000<BR>\n" );
document.write( "t ≈ 10.83\r<BR>\n" );
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document.write( "Since t represents the number of years after 1981, we add this to 1981:\r<BR>\n" );
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document.write( "1981 + 10.83 ≈ 1991.83\r<BR>\n" );
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document.write( "Since we're dealing with whole numbers of years, ridership will likely fall below 180,000 in the year **1991** or **1992**.<BR>\n" );
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