document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=201783>Question 201783</A>: <I><SPAN STYLE=\"word-break: break-all;\"> strontium-90 is a waste product from nuclear fission reactors. It's half-life is 28 years and its decay model as A=A(subscript 0)e^-0.024755t. Suppose a nuclear accident occurs and releases 50 grams of strontium-90 into the biosphere. How long will it take for strontium-90 to decay to a level of 20 grams?  I don't have a clue how to work this problem, can someone help? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #152027 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=152027\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In general, if you have values for all but one of the variables in an equation, you can use the equation to find the last variable. In your equation you have three variables: A, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B0%5D ALIGN=MIDDLE  ALT=\"A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and t. And you only know what you want A to be: 20. We want to find the time (t) so we need to know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B0%5D ALIGN=MIDDLE  ALT=\"A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > first.<br><BR>\n" );
document.write( "To find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B0%5D ALIGN=MIDDLE  ALT=\"A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, think about what happens when t=0. If t=0 then the exponent becomes 0. And what is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E0 ALIGN=MIDDLE  ALT=\"e%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >? Any number (except 0) raised to the 0 power is 1! So when t=0 your equation becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+A%5B0%5D ALIGN=MIDDLE  ALT=\"A+=+A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B0%5D ALIGN=MIDDLE  ALT=\"A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the starting (t=0) amount! (Tip: This is what subscripts of zero often represent: a starting value for the variable with the same name without the subscript.)<br><BR>\n" );
document.write( "Since we now know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B0%5D ALIGN=MIDDLE  ALT=\"A%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the starting amount of strontium-90 and we want to solve a problem for a starting amount of 50, we now have an equation we can use to find the answer:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20+=+50e%5E%28-0.024755t%29 ALIGN=MIDDLE  ALT=\"20+=+50e%5E%28-0.024755t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We will now solve this for t. Start by dividing both sides by 50:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.4+=+e%5E%28-0.024755t%29 ALIGN=MIDDLE  ALT=\"0.4+=+e%5E%28-0.024755t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now find the natural logarithm (ln) of both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%280.4%29+=+ln%28e%5E%28-0.024755t%29%29 ALIGN=MIDDLE  ALT=\"ln%280.4%29+=+ln%28e%5E%28-0.024755t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The left side we can get from our calculator. The right side, if you understand logarithms well, simplifies easily. This is why we used ln to begin with. It is an easy way to get a variable out of an exponent. And we can't solve for t until it is out of the exponent.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.9162907318741551+=+-0.024755t ALIGN=MIDDLE  ALT=\"-0.9162907318741551+=+-0.024755t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we just divide both sides by -0.024755:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37.0143701019654642+=+t ALIGN=MIDDLE  ALT=\"37.0143701019654642+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This tells us that it will take a little more than 37 years for 50g of strontium-90 to decay to 20g. </SPAN>\n" );
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