document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1101289>Question 1101289</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The count in a culture of bacteria was 200 after 2 hours and 25000 after 5 hours.<BR>\n" );
document.write( "a. Find a function that models the number of bacteria n(t) after t hours.<BR>\n" );
document.write( "b. When does the culture contain 2,000 bacteria?\r<BR>\n" );
document.write( "\n" );
document.write( "I am taking an online pre-calc math class. Our math teacher does not make videos for us but I have a text book. However, I feel like the text book does not have enough examples for us and I need help on this problem. I would appreciate to see the work so I can understand the problem. Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #715895 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=715895\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The number of bacteria is increasing exponentially, so the function is of the form<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;\" ><BR>\n" );
document.write( "where a is an initial value (when x=0) and b is the growth rate.<br><BR>\n" );
document.write( "(1) Use the two given pieces of information to write equations in that form, using x as the time in hours:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200+=+ab%5E2 ALIGN=MIDDLE  ALT=\"200+=+ab%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25000+=+ab%5E5 ALIGN=MIDDLE  ALT=\"25000+=+ab%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) Divide the second equation by the first; that will eliminate a, allowing you to find the value of b:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125+=+b%5E3 ALIGN=MIDDLE  ALT=\"125+=+b%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+=+b ALIGN=MIDDLE  ALT=\"5+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Remember not to just go through the motions of solving the problem; take the time to realize that b=5 means the number of bacteria grows by a factor of 5 each hour.<BR>\n" );
document.write( "(You will appreciate the mathematics more if you understand what it is doing....)<br><BR>\n" );
document.write( "(3) Use the value you have found for b in either of the original equations to find the value of a:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200+=+a%285%29%5E2 ALIGN=MIDDLE  ALT=\"200+=+a%285%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200+=+25a ALIGN=MIDDLE  ALT=\"200+=+25a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+8 ALIGN=MIDDLE  ALT=\"a+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+8%285%29%5Ex ALIGN=MIDDLE  ALT=\"y+=+8%285%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For part b, you need to solve the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%285%29%5Ex+=+2000 ALIGN=MIDDLE  ALT=\"8%285%29%5Ex+=+2000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Up to now, the numbers have worked out nicely; a and b are whole numbers.  That will no longer be the case:<BR>\n" );
document.write( "f(0) = 8<BR>\n" );
document.write( "f(1) = 40<BR>\n" );
document.write( "f(2) = 200<BR>\n" );
document.write( "f(3) = 1000<BR>\n" );
document.write( "f(4) = 5000<BR>\n" );
document.write( "f(5) = 25000<BR>\n" );
document.write( "So the solution to part b will be between 3 and 4 hours.<br><BR>\n" );
document.write( "To solve algebraically, because the variable is in an exponent, you need to use logarithms.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%285%5Ex%29+=+2000 ALIGN=MIDDLE  ALT=\"8%285%5Ex%29+=+2000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex+=+250 ALIGN=MIDDLE  ALT=\"5%5Ex+=+250\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Alog%285%29+=+log%28250%29 ALIGN=MIDDLE  ALT=\"x%2Alog%285%29+=+log%28250%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%28250%29%2Flog%285%29+=+3.43 ALIGN=MIDDLE  ALT=\"x+=+log%28250%29%2Flog%285%29+=+3.43\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (to 2 decimal places) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );