document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204141>Question 204141</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could anyone help me with this. I am not sure how to do it. I really need your help.\r<BR>\n" );
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document.write( "Do exponential functions only model phenomena that grow, or can they also model phenomena that decay?  Explain what is different in the form of the function in each case. <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 #154135 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=154135\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Exponential functions model both phenomena that grow and phenomena that decay.<BR>\n" );
document.write( "The general form for the exponential function is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ab%5Ex ALIGN=MIDDLE  ALT=\"y+=+ab%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > b is called the base.<BR>\n" );
document.write( "For growth phenomena, the base, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E+1 ALIGN=MIDDLE  ALT=\"b+%3E+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For decay phenomena, the base, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+b+%3C+1 ALIGN=MIDDLE  ALT=\"0+%3C+b+%3C+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A  typical example of exponential grow is the increase in money deposited in savings account.<BR>\n" );
document.write( "For example, if you deposited $500 at 5% interest per year, how much would you have at the end of 4 years?<BR>\n" );
document.write( "The formula is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+P%281%2Br%29%5Et ALIGN=MIDDLE  ALT=\"A+=+P%281%2Br%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > where:<BR>\n" );
document.write( "A = the amount you would have at the end of 4 years..<BR>\n" );
document.write( "P = the principal (amount invested).<BR>\n" );
document.write( "r = the rate of interest, in decimal form.<BR>\n" );
document.write( "t = the length of time deposited, in years.<BR>\n" );
document.write( "In our example, P = $500, r = 0.05, and t = 4 years.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+500%281%2B0.05%29%5E4 ALIGN=MIDDLE  ALT=\"A+=+500%281%2B0.05%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+500%281.05%29%5E4 ALIGN=MIDDLE  ALT=\"A+=+500%281.05%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use your calculator to get the approximate value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.05%29%5E4 ALIGN=MIDDLE  ALT=\"1.05%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+607.75 ALIGN=MIDDLE  ALT=\"A+=+607.75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You would have $607.75 in 4 years.<BR>\n" );
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document.write( "A typical example of exponential decay is the decrease of radio-active material (radio-active decay) after a length of time.<BR>\n" );
document.write( "The term \"half-life\" is used to indicate the length of time it takes for radio-active material to lose half of its mass.<BR>\n" );
document.write( "For example:<BR>\n" );
document.write( "The half-life of an isotope of thorium, thorium-234, is 25 days.<BR>\n" );
document.write( "If you started with 50 grams of thorium- 234, how much would be left after 100 days?<BR>\n" );
document.write( "Since the amount of thorium-234 decreases 50% every 25 days, the exponential function for the decay is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+50%280.5%29%5Et ALIGN=MIDDLE  ALT=\"y+=+50%280.5%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > where t = the number of half-lives that have elapsed. Notice that the base (b = 0.5) is less than 1.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+100%2F25 ALIGN=MIDDLE  ALT=\"t+=+100%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+4 ALIGN=MIDDLE  ALT=\"t+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >half-lives.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+50%280.5%29%5E4 ALIGN=MIDDLE  ALT=\"y+=+50%280.5%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+50%280.0625%29 ALIGN=MIDDLE  ALT=\"y+=+50%280.0625%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3.125 ALIGN=MIDDLE  ALT=\"y+=+3.125\"   DISABLED_style= \"max-width:90%; height:auto;\" >grams. </SPAN>\n" );
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