document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163697>Question 1163697</A>: <I><SPAN STYLE=\"word-break: break-all;\"> unsolved	2020-08-22 10:07:30	It is believed that two quantities, z and d are Connected by the relationship of the form z=kd^n where k and n are provided that d doesn't exceed some fixed (but unknown)values D.An experiment produced the following data<BR>\n" );
document.write( "D 750 810 870 930 990 1050 1110 1170<BR>\n" );
document.write( "Z 2.1 2.6 3.2 4.0 4.8 5.6 5.9 6.1<BR>\n" );
document.write( "a)Plot the values of log10Z against log10 d.Use these points to suggest a<BR>\n" );
document.write( "value for D.<BR>\n" );
document.write( "b)It is known tht for d < D,n is a whole number.Use your graph to find the value of n.Show also that k=5×10^-9<BR>\n" );
document.write( "c) Use your value of n and the estimate k=5×10^-9 to find the value of d for which z=3.0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853915 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5342)</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=853915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=k%2Ad%5En ALIGN=MIDDLE  ALT=\"z=k%2Ad%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cz%29=log%2810%2Ck%29%2Bn%2Alog%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cz%29=log%2810%2Ck%29%2Bn%2Alog%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cz%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cz%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > against <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be expected to be a straight line with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > up to a certain value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cd%29=log%2810%2CD%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cd%29=log%2810%2CD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We need to calculate and tabulate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cz%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cz%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%289%2C4%2Cd%2Cz%2Clog%2810%2Cd%29%2Clog%2810%2Cz%29%2C750%2C2.1%2C2.875061263%2C0.322219295%0D%0A%2C810%2C2.6%2C2.908485019%2C0.414973348%2C870%2C3.2%2C2.939519253%2C0.505149978%2C930%2C4.0%2C2.968482949%2C0.602059991%2C990%2C4.8%2C2.995635195%2C0.681241237%2C1050%2C5.6%2C3.021189299%2C0.748188027%2C1110%2C5.9%2C3.045322979%2C0.770852012%2C1170%2C6.1%2C3.068185862%2C0.785329835%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >    <BR>\n" );
document.write( " <BR>\n" );
document.write( "a)Then we plot <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cz%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cz%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > against <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+300%2C+300%2C+2.8%2C+3.1%2C0.25%2C0.85%2C%0D%0Agraph%28+300%2C+300%2C+2.8%2C+3.1%2C0.25%2C0.85%2C0.6332x-1.1576%2C2.9634x-8.2005%29%2C%0D%0Aarrow%282.81%2C0.55%2C2.81%2C0.2%29%2Carrow%282.81%2C0.55%2C2.81%2C0.85%2C0.45%29%2C%0D%0Aline%282.81%2C0.35%2C2.805%2C0.35%29%2Cline%282.81%2C0.45%2C2.805%2C0.45%29%2C%0D%0Aline%282.81%2C0.55%2C2.805%2C0.55%29%2Cline%282.81%2C0.65%2C2.805%2C0.65%29%2C%0D%0Aline%282.81%2C0.75%2C2.805%2C0.75%29%2Clocate%282.815%2C0.77%2C0.75%29%2C%0D%0Alocate%282.815%2C0.67%2C0.65%29%2Clocate%282.815%2C0.57%2C0.55%29%2C%0D%0Alocate%282.815%2C0.47%2C0.45%29%2Clocate%282.815%2C0.37%2C0.35%29%2C%0D%0Aarrow%282.95%2C0.27%2C2.7%2C0.27%29%2Carrow%282.95%2C0.27%2C3.1%2C0.27%29%2C%0D%0Aline%282.85%2C0.275%2C2.85%2C0.265%29%2Cline%282.9%2C0.275%2C2.9%2C0.265%29%2C%0D%0Aline%282.95%2C0.275%2C2.95%2C0.265%29%2Cline%283%2C0.275%2C3%2C0.265%29%2C%0D%0Aline%283.05%2C0.275%2C3.05%2C0.265%29%2Clocate%282.835%2C0.305%2C2.85%29%2C%0D%0Alocate%282.885%2C0.305%2C2.90%29%2Clocate%282.935%2C0.305%2C2.95%29%2C%0D%0Alocate%282.985%2C0.305%2C%223.00%22%29%2Clocate%283.035%2C0.305%2C3.05%29%2C%0D%0Agreen%28circle%282.875%2C0.3222%2C0.005%29%29%2Cgreen%28circle%282.908%2C0.415%2C0.005%29%29%2C%0D%0Agreen%28circle%282.94%2C0.5051%2C0.005%29%29%2Cgreen%28circle%282.968%2C0.6021%2C0.005%29%29%2C%0D%0Agreen%28circle%282.996%2C0.6812%2C0.005%29%29%2Cgreen%28circle%283.021%2C0.7482%2C0.005%29%29%2C%0D%0Ared%28circle%283.045%2C0.7709%2C0.005%29%29%2Cred%28circle%283.068%2C0.7853%2C0.005%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The points in green, up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cd%29=3.021 ALIGN=MIDDLE  ALT=\"log%2810%2Cd%29=3.021\"   DISABLED_style= \"max-width:90%; height:auto;\" > , corresponding to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=1050 ALIGN=MIDDLE  ALT=\"d=1050\"   DISABLED_style= \"max-width:90%; height:auto;\" > fit well enough on a line. That supports suggesting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28D=1050%29 ALIGN=MIDDLE  ALT=\"highlight%28D=1050%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "b) We could estimate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the slope between points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cz%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cz%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22 ALIGN=MIDDLE  ALT=\"%22%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.875 ALIGN=MIDDLE  ALT=\"2.875\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.3222 ALIGN=MIDDLE  ALT=\"0.3222\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22 ALIGN=MIDDLE  ALT=\"%22%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.021 ALIGN=MIDDLE  ALT=\"3.021\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.7482 ALIGN=MIDDLE  ALT=\"0.7482\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22 ALIGN=MIDDLE  ALT=\"%22%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280.7482-0.3222%29%2F%283.021-2.875%29=0.4260%2F0.146=2.92%28%22rounded%22%29 ALIGN=MIDDLE  ALT=\"%280.7482-0.3222%29%2F%283.021-2.875%29=0.4260%2F0.146=2.92%28%22rounded%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is supposed to be a whole number, it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28n=3%29 ALIGN=MIDDLE  ALT=\"highlight%28n=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "c) For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=5%C3%9710%5E-9 ALIGN=MIDDLE  ALT=\"k=5%C3%9710%5E-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Ck%29=log%2810%2C5%2A10%5E-9%29=-8.30103 ALIGN=MIDDLE  ALT=\"log%2810%2Ck%29=log%2810%2C5%2A10%5E-9%29=-8.30103\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)\r<BR>\n" );
document.write( "\n" );
document.write( "Substituting values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=3.0 ALIGN=MIDDLE  ALT=\"z=3.0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=5%C3%9710%5E-9 ALIGN=MIDDLE  ALT=\"k=5%C3%9710%5E-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Ck%29=-8.30103%0D%0A ALIGN=MIDDLE  ALT=\"log%2810%2Ck%29=-8.30103%0D%0A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C3%29=0.47712 ALIGN=MIDDLE  ALT=\"log%2810%2C3%29=0.47712\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cz%29=log%2810%2Ck%29%2Bn%2Alog%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cz%29=log%2810%2Ck%29%2Bn%2Alog%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.47712=-8.30103%2B3log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"0.47712=-8.30103%2B3log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.47712%2B8.30103=3log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"0.47712%2B8.30103=3log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8.77815=3log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"8.77815=3log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8.77815%2F3=log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"8.77815%2F3=log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.92615=log%2810%2Cd%29 ALIGN=MIDDLE  ALT=\"2.92615=log%2810%2Cd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=10%5E2.92615 ALIGN=MIDDLE  ALT=\"d=10%5E2.92615\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28d=843%29 ALIGN=MIDDLE  ALT=\"highlight%28d=843%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the value of d for which z=3.0. </SPAN>\n" );
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