document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=239293>Question 239293</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When light strikes the surface of a medium such as water or glass, its intensity decreases with depth. The beer-lambert-bougar law states that the percentage of decrease is the same for each additional unit of depth. In a certain lake, intensity decreases about 75% for each additionial meter of depth.\r<BR>\n" );
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document.write( "a) Explain why intensity I is an exponential function of depth d in meters<BR>\n" );
document.write( "b) Use a formula to express intensity I as an exponential function of d. (use I0 to denote the initial intensity.)<BR>\n" );
document.write( "c) Explain in practical terms the meaning of I0.<BR>\n" );
document.write( "d) At what depth will the intensity of light be one tenth of the intensity of light striking the surface? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #175754 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=175754\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Some keys to understanding how to do this problem:<ul><li>Using percents in formulas or functions is not very practical. So, since 75% is 3/4, we will use 3/4 instead of 75%. (We could also use 0.75 instead of 75%.)</li><li>When problems say \"75% of\" some number (or \"3/4 of\") some number, that \"of\" translates into a multiplication. The beer-lambert-bougar law says that the intensity of light is some percent of the intensity of the light 1 meter above. So to find the intensity of light in your lake, we will multiply the intensity of light above by 75% (or 3/4).</li></ul><BR>\n" );
document.write( "So to figure out the answers to your problem, let's make up a number for the intensity of light just as it hits the water (i.e. the depth is zero):<ul><li> Let's say the intensity at depth 0 is 100.</li><li>The intensity at a depth of 1 meter would be 3/4 of intensity of the light at 0 meters: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%283%2F4%29 ALIGN=MIDDLE  ALT=\"100%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.</li><li>At 2 meters depth the intensity would be 3/4 of the intensity at 1 meters depth: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%283%2F4%29%283%2F4%29 ALIGN=MIDDLE  ALT=\"100%283%2F4%29%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>At 3 meters depth the intensity would be 3/4 of the intensity at 2 meters depth: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%283%2F4%29%283%2F4%29%283%2F4%29 ALIGN=MIDDLE  ALT=\"100%283%2F4%29%283%2F4%29%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>etc.</li></ul><BR>\n" );
document.write( "a) Explain why intensity I is an exponential function of depth d in meters<BR>\n" );
document.write( "We can see that the intensity is calculated, in part, by the repeated multiplication of 3/4. Our formula is an exponential function because we will use an exponent to represent the repeated multiplication of 3/4.<br><BR>\n" );
document.write( "b) Use a formula to express intensity I as an exponential function of d. (use I0 to denote the initial intensity.)<BR>\n" );
document.write( "From the example above we can see where the initial intensity (intensity at depth 0) fits in the formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I+=+I%5B0%5D%283%2F4%29%5Ed ALIGN=MIDDLE  ALT=\"I+=+I%5B0%5D%283%2F4%29%5Ed\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where I is the intensity at d meters, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I%5B0%5D ALIGN=MIDDLE  ALT=\"I%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial intensity and d is the depth in meters.<br><BR>\n" );
document.write( "c) Explain in practical terms the meaning of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I%5B0%5D ALIGN=MIDDLE  ALT=\"I%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "It is the intensity at depth 0 meters<br><BR>\n" );
document.write( "d) At what depth will the intensity of light be one tenth of the intensity of light striking the surface?<BR>\n" );
document.write( "One tenth of the intensity of light striking the surface is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F10%29I%5B0%5D ALIGN=MIDDLE  ALT=\"%281%2F10%29I%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (Remember that \"a fraction of a number\" means that fraction times that number.) So the equation we need to solve is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F10%29I%5B0%5D+=+I%5B0%5D%283%2F4%29%5Ed ALIGN=MIDDLE  ALT=\"%281%2F10%29I%5B0%5D+=+I%5B0%5D%283%2F4%29%5Ed\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "First we'll divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I%5B0%5D ALIGN=MIDDLE  ALT=\"I%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2F10%29I%5B0%5D%29%2FI%5B0%5D+=+%28I%5B0%5D%283%2F4%29%5Ed%29%2FI%5B0%5D ALIGN=MIDDLE  ALT=\"%28%281%2F10%29I%5B0%5D%29%2FI%5B0%5D+=+%28I%5B0%5D%283%2F4%29%5Ed%29%2FI%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "On both sides the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I%5B0%5D ALIGN=MIDDLE  ALT=\"I%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s cancel leaving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F10+=+%283%2F4%29%5Ed ALIGN=MIDDLE  ALT=\"1%2F10+=+%283%2F4%29%5Ed\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To solve for d, when it is the exponent like this, we will use logarithms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%281%2F10%29%29+=+log%28%28%283%2F4%29%5Ed%29%29 ALIGN=MIDDLE  ALT=\"log%28%281%2F10%29%29+=+log%28%28%283%2F4%29%5Ed%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The left side is -1 because 1/10 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E-1 ALIGN=MIDDLE  ALT=\"10%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. (If this is not obvious, use your calculator. So now we have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+log%28%28%283%2F4%29%5Ed%29%29 ALIGN=MIDDLE  ALT=\"-1+=+log%28%28%283%2F4%29%5Ed%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the right side we can use the property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to move the exponent from the argument out in front of the logarithm giving us:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+d%2Alog%28%283%2F4%29%29 ALIGN=MIDDLE  ALT=\"-1+=+d%2Alog%28%283%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And now we can divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%283%2F4%29%29 ALIGN=MIDDLE  ALT=\"log%28%283%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%29%2Flog%28%283%2F4%29%29%29+=+%28%28d%2Alog%28%283%2F4%29%29%29%2Flog%28%283%2F4%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the right side the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%283%2F4%29%29 ALIGN=MIDDLE  ALT=\"log%28%283%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s cancel leaving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%29%2Flog%28%283%2F4%29%29%29+=+d ALIGN=MIDDLE  ALT=\"%28%28-1%29%2Flog%28%283%2F4%29%29%29+=+d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using our calculator on the remaining log:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%29%2F-0.1249387366083000%29+=+d ALIGN=MIDDLE  ALT=\"%28%28-1%29%2F-0.1249387366083000%29+=+d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And using the calculator to divide we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8.0039227796510935+=+d ALIGN=MIDDLE  ALT=\"8.0039227796510935+=+d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So at very close to 8 meters the intensity of light will be 1/10 of what it was at the surface.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );