document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186430>Question 1186430</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A car was valued at $27,000 in the year 1994. The value depreciated to $12,000 by the year 2001.<BR>\n" );
document.write( "A) What was the annual rate of change between 1994 and 2001?<BR>\n" );
document.write( "r<BR>\n" );
document.write( "  = <BR>\n" );
document.write( " Round the rate of decrease to 4 decimal places.\r<BR>\n" );
document.write( "\n" );
document.write( "B) What is the correct answer to part A written in percentage form?<BR>\n" );
document.write( "r<BR>\n" );
document.write( "  = <BR>\n" );
document.write( " %.\r<BR>\n" );
document.write( "\n" );
document.write( "C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2006 ?<BR>\n" );
document.write( "value = $ <BR>\n" );
document.write( " Round to the nearest 50 dollars.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849443 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=849443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A) To find the annual rate of change, we can use the formula for exponential decay:\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "Final Value = Initial Value * (1 - r)^t<BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "Where:\r<BR>\n" );
document.write( "\n" );
document.write( "* Final Value = $12,000<BR>\n" );
document.write( "* Initial Value = $27,000<BR>\n" );
document.write( "* r = annual rate of change (what we want to find)<BR>\n" );
document.write( "* t = number of years = 2001 - 1994 = 7\r<BR>\n" );
document.write( "\n" );
document.write( "Let's plug in the values and solve for r:\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "12000 = 27000 * (1 - r)^7<BR>\n" );
document.write( "(12000/27000) = (1 - r)^7<BR>\n" );
document.write( "(4/9) = (1 - r)^7<BR>\n" );
document.write( "(4/9)^(1/7) = 1 - r<BR>\n" );
document.write( "r = 1 - (4/9)^(1/7)<BR>\n" );
document.write( "r ≈ 0.1094<BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the annual rate of change is approximately **0.1094**.\r<BR>\n" );
document.write( "\n" );
document.write( "B) To express the rate in percentage form, we simply multiply by 100:\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "r ≈ 0.1094 * 100 = 10.94%<BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the annual rate of change is approximately **10.94%**.\r<BR>\n" );
document.write( "\n" );
document.write( "C) To find the value in 2006, we can use the same formula, but with t = 2006 - 1994 = 12:\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "Value in 2006 = 27000 * (1 - 0.1094)^12<BR>\n" );
document.write( "Value in 2006 ≈ 6715.64<BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "Rounding to the nearest 50 dollars, the value of the car in 2006 will be approximately **$6700**. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );