document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=998886>Question 998886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help and show all the steps.\r<BR>\n" );
document.write( "\n" );
document.write( "The function C(t)=5t/0.01t^2+3.3 describes the concentration of a drug in the blood stream over time when take orally. C is measured in micrograms per milliliter and t is measured in minutes.\r<BR>\n" );
document.write( "\n" );
document.write( "A). Sketch a graph of the function over the first two hours after the dose is given. Label the axes.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "B). Determine when the maximum amount of the drug is in the body and the concentration at that time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "C). Explain within the context of the problem the shape of the graph between taking the drug orally(t=0)and the maximum point. What does the shape of the graph tell us happens between the maximum point and two hours after taking the drug?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "D.) Does this function have any asymptotes? If so, where are they? What do the asymptotes tell us about the concentration of a drug in our blood stream? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #616625 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=616625\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The function should be C(t)=5t/(0.01t^2+3.3) =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5t%2F%280.01t%5E2%2B3.3%29 ALIGN=MIDDLE  ALT=\"5t%2F%280.01t%5E2%2B3.3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because 5t/0.01t^2+3.3 =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5t%2F0.01t%5E2 ALIGN=MIDDLE  ALT=\"5t%2F0.01t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.3 ALIGN=MIDDLE  ALT=\"3.3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that would mean infinite concentration at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which does not make sense.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A) The graph for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28t%29=5t%2F%280.01t%5E2%2B3.3%29 ALIGN=MIDDLE  ALT=\"C%28t%29=5t%2F%280.01t%5E2%2B3.3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-50%2C450%2C-2%2C18%2C%0D%0Agraph%28300%2C300%2C-50%2C450%2C-2%2C18%2C5x%2F%280.01x%5E2%2B3.3%29%29%2C%0D%0Alocate%2810%2C18%2C%22C+%2Cmcg%2FmL%22%29%2Clocate%28340%2C-1%2C%22t+%2C+minutes%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph above is not a sketch, but a more precise graph, such as we would get with a graphing calculator, otr some software with f-graphing capabilities.\r<BR>\n" );
document.write( "\n" );
document.write( "B) Calculus tells us that the function will have a maximum, when the derivative <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2Fdt ALIGN=MIDDLE  ALT=\"dC%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > is zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2Fdt=%285%280.01t%5E2%2B3.3%29-5t%280.01%2A2t%29%29%2F%280.01t%5E2%2B3.3%29%5E2=%280.05t%5E2%2B16.5-0.1t%5E2%29%2F%280.01t%5E2%2B3.3%29%5E2=%28-0.05t%5E2%2B16.5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2Fdt=0 ALIGN=MIDDLE  ALT=\"dC%2Fdt=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.05t%5BMAX%5D%5E2%2B16.5=0 ALIGN=MIDDLE  ALT=\"-0.05t%5BMAX%5D%5E2%2B16.5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5BMAX%5D=sqrt%2816.5%2F0.05%29=sqrt%28330%29=about18.2minutes ALIGN=MIDDLE  ALT=\"t%5BMAX%5D=sqrt%2816.5%2F0.05%29=sqrt%28330%29=about18.2minutes\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3Ct%5BMAX%5D ALIGN=MIDDLE  ALT=\"t%3Ct%5BMAX%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , {{dC/dt>0, meaning that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%28t%29 ALIGN=MIDDLE  ALT=\"c%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is increasing.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3Et%5BMAX%5D ALIGN=MIDDLE  ALT=\"t%3Et%5BMAX%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , {{dC/dt<0, meaning that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%28t%29 ALIGN=MIDDLE  ALT=\"c%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is decreasing.<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=t%5BMAX%5D ALIGN=MIDDLE  ALT=\"t=t%5BMAX%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28t%5BMAX%5D%29=C%5BMAX%5D=5sqrt%28330%29%2F%280.01%2A330%2B3.3%29=5sqrt%28330%29%2F6.6=about13.8 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22mcg%2FmL%22 ALIGN=MIDDLE  ALT=\"%22mcg%2FmL%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C45%2C-2%2C18%2C%0D%0Agraph%28300%2C300%2C-5%2C45%2C-2%2C18%2C5x%2F%280.01x%5E2%2B3.3%29%29%2C%0D%0Ared%28circle%2818.2%2C13.8%2C0.5%29%29%2C+locate%2818%2C16%2Cred%28C%5BMAX%5D%29%29%2C%0D%0Alocate%281%2C18%2C%22C+%2Cmcg%2FmL%22%29%2Clocate%2834%2C-1%2C%22t+%2C+minutes%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "C) Between taking the drug orally (t=0) and the maximum point the blood concentration increases rapidly at first, and the absorption rate decreases towards the maximum. (We see the changes in absorption rate as changes of the slope on the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28t%29 ALIGN=MIDDLE  ALT=\"C%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "After the early <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5BMAX%5D=18.2minutes ALIGN=MIDDLE  ALT=\"t%5BMAX%5D=18.2minutes\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the concentration of the drug in the blood decreases, slowly at first, followed by a faster rate of decrease, and the s slower rate of decrease towards <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.1 ALIGN=MIDDLE  ALT=\"4.1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22mcg%2FmL%22 ALIGN=MIDDLE  ALT=\"%22mcg%2FmL%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > at two hours.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2820minutes%29=5%2A20%2F%280.01%2A400%2B3.3%29=100%2F7.3=about+13.7 ALIGN=MIDDLE  ALT=\"C%2820minutes%29=5%2A20%2F%280.01%2A400%2B3.3%29=100%2F7.3=about+13.7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22mcg%2FmL%22 ALIGN=MIDDLE  ALT=\"%22mcg%2FmL%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2830minutes%29=5%2A30%2F%280.01%2A900%2B3.3%29=150%2F12.3=about+12.2 ALIGN=MIDDLE  ALT=\"C%2830minutes%29=5%2A30%2F%280.01%2A900%2B3.3%29=150%2F12.3=about+12.2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22mcg%2FmL%22 ALIGN=MIDDLE  ALT=\"%22mcg%2FmL%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2860minutes%29=5%2A60%2F%280.01%2A3600%2B3.3%29=300%2F39.3=about+7.6 ALIGN=MIDDLE  ALT=\"C%2860minutes%29=5%2A60%2F%280.01%2A3600%2B3.3%29=300%2F39.3=about+7.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22mcg%2FmL%22 ALIGN=MIDDLE  ALT=\"%22mcg%2FmL%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%282hours%29=C%28120minutes%29=5%2A120%2F%280.01%2A14400%2B3.3%29=600%2F147.3=about4.1 ALIGN=MIDDLE  ALT=\"C%282hours%29=C%28120minutes%29=5%2A120%2F%280.01%2A14400%2B3.3%29=600%2F147.3=about4.1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22mcg%2FmL%22 ALIGN=MIDDLE  ALT=\"%22mcg%2FmL%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "IMHO:<BR>\n" );
document.write( "The drug must start being absorbed in the stomach (if not the buccal mucosa), because the rate of increase in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28t%29 ALIGN=MIDDLE  ALT=\"C%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is maximal at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and if it were not substantially absorbed before the intestine,<BR>\n" );
document.write( "we would see a lower rate at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , followed by increasing absorption over some time,<BR>\n" );
document.write( "before seeing the absorption rate decrease towards <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5BMAX%5D ALIGN=MIDDLE  ALT=\"C%5BMAX%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > happening at a longer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5BMAX%5D ALIGN=MIDDLE  ALT=\"t%5BMAX%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "D) The function has only an horizontal asymptote: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=0 ALIGN=MIDDLE  ALT=\"C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28t%29%3E0 ALIGN=MIDDLE  ALT=\"C%28t%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches zero asymptotically as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases.<BR>\n" );
document.write( "That tells us that the concentration in the blood decreases towards zero,<BR>\n" );
document.write( "but theoretically there is some minute (and decreasing) level of the drug in the blood forever. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );