document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127357>Question 127357</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321227573/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Please help, I'm stuck!<BR>\n" );
document.write( "(Modeling)<BR>\n" );
document.write( "In electronics, circuit gain is modeled by\r<BR>\n" );
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document.write( "G = Bt/R + R(base)t\r<BR>\n" );
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document.write( "Where R is the value of the resistor, t is temperture, R(base)t is the value of R at room temperture t and B is constant. The sensitivity of the circuit to temperture is modeled by\r<BR>\n" );
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document.write( "S = BR/(R + R (base)t)^2\r<BR>\n" );
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document.write( "If B = 3.7 and t is 90 K, find the values of R and R (base)t that will make G = .4 and S = .001.\r<BR>\n" );
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document.write( "I huge thank you to anyone who can get me goingon the right track. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93330 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=93330\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I hope that I correctly interpret your two given equations (one for gain and the other for<BR>\n" );
document.write( "sensitivity) as being:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G+=+%28B%2At%29%2F%28R+%2B+R%5Bt%5D%29 ALIGN=MIDDLE  ALT=\"G+=+%28B%2At%29%2F%28R+%2B+R%5Bt%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "For this one I'm guessing on what the right side of this equation should be. You wrote it as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G+=+%28B%2At%29%2FR+%2B+R%5Bt%5D ALIGN=MIDDLE  ALT=\"G+=+%28B%2At%29%2FR+%2B+R%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "but I'm guessing that my version above is you intended. [If I'm wrong, you may want to repost your <BR>\n" );
document.write( "problem and we'll try again.]<BR>\n" );
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document.write( "and <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28B%2AR%29%2F%28R+%2B+R%5Bt%5D%29%5E2 ALIGN=MIDDLE  ALT=\"S+=+%28B%2AR%29%2F%28R+%2B+R%5Bt%5D%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You are given that B = 3.7, t = 90, G = 0.4, S = 0.001<BR>\n" );
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document.write( "Substitute these values into the two equations and they become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.4+=+%283.7%2A90%29%2F%28R+%2B+R%5Bt%5D%29 ALIGN=MIDDLE  ALT=\"0.4+=+%283.7%2A90%29%2F%28R+%2B+R%5Bt%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== call this equation #1<BR>\n" );
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document.write( "and<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.001+=+%283.7%2AR%29%2F%28R+%2B+R%5Bt%5D%29%5E2 ALIGN=MIDDLE  ALT=\"0.001+=+%283.7%2AR%29%2F%28R+%2B+R%5Bt%5D%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== call this equation #2<BR>\n" );
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document.write( "Go to equation #1 and multiply both sides by the quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+%2B+R%5Bt%5D ALIGN=MIDDLE  ALT=\"R+%2B+R%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When you do,<BR>\n" );
document.write( "on the right side this multiplication cancels the denominator and you are left with equation #1 <BR>\n" );
document.write( "becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.4%2A%28R+%2B+R%5Bt%5D%29+=+3.7%2A90 ALIGN=MIDDLE  ALT=\"0.4%2A%28R+%2B+R%5Bt%5D%29+=+3.7%2A90\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the right side and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.4%2A%28R+%2B+R%5Bt%5D%29+=+333 ALIGN=MIDDLE  ALT=\"0.4%2A%28R+%2B+R%5Bt%5D%29+=+333\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+%2B+R%5Bt%5D ALIGN=MIDDLE  ALT=\"R+%2B+R%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by dividing both sides of this equation by 0.4 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+%2B+R%5Bt%5D+=+333%2F0.4+=+832.5 ALIGN=MIDDLE  ALT=\"R+%2B+R%5Bt%5D+=+333%2F0.4+=+832.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You now have a numerical value of 832.5 ohms for R + R[t] and you can substitute that value in<BR>\n" );
document.write( "for the denominator in equation #2 to make equation #2 become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.001+=+%283.7%2AR%29%2F%28832.5%29%5E2 ALIGN=MIDDLE  ALT=\"0.001+=+%283.7%2AR%29%2F%28832.5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Square out the denominator on the right side and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.001+=+%283.7%2AR%29%2F693056.25 ALIGN=MIDDLE  ALT=\"0.001+=+%283.7%2AR%29%2F693056.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the denominator on the right side by multiplying both sides of this equation by<BR>\n" );
document.write( "693056.25 and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.001%2A693056.25+=+3.7%2AR ALIGN=MIDDLE  ALT=\"0.001%2A693056.25+=+3.7%2AR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the left side and you reduce the equation to <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=693.05625+=+3.7%2AR ALIGN=MIDDLE  ALT=\"693.05625+=+3.7%2AR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally solve for R by dividing both sides by 3.7 and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=187.3125+=+R ALIGN=MIDDLE  ALT=\"187.3125+=+R\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "[If only resistors could be manufactured to that precision ...]<BR>\n" );
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document.write( "Now that we have a value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can return to either of the original equations  and substitute<BR>\n" );
document.write( "this value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve the resulting equation for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5Bt%5D ALIGN=MIDDLE  ALT=\"R%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Let's return to equation #1 and set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+=+187.3125 ALIGN=MIDDLE  ALT=\"R+=+187.3125\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This makes equation #1 become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.4+=+%283.7%2A90%29%2F%28187.3125+%2B+R%5Bt%5D%29 ALIGN=MIDDLE  ALT=\"0.4+=+%283.7%2A90%29%2F%28187.3125+%2B+R%5Bt%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Again multiply out the numerator on the right side and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.4+=+333%2F%28187.3125+%2B+R%5Bt%5D%29 ALIGN=MIDDLE  ALT=\"0.4+=+333%2F%28187.3125+%2B+R%5Bt%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the denominator on the right side of this equation by multiplying both sides<BR>\n" );
document.write( "by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=187.3125+%2B+R%5Bt%5D ALIGN=MIDDLE  ALT=\"187.3125+%2B+R%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the equation then is changed to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.4%28187.3125+%2B+R%5Bt%5D%29+=+333 ALIGN=MIDDLE  ALT=\"0.4%28187.3125+%2B+R%5Bt%5D%29+=+333\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the multiplier 0.4 by dividing both sides of the equation by 0.4 and the equation<BR>\n" );
document.write( "is reduced to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=187.3125+%2B+R%5Bt%5D+=+832.5 ALIGN=MIDDLE  ALT=\"187.3125+%2B+R%5Bt%5D+=+832.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the constant on the left side by subtracting 187.3125 from both sides and you are <BR>\n" );
document.write( "left with the answer of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5Bt%5D+=+645.1875 ALIGN=MIDDLE  ALT=\"R%5Bt%5D+=+645.1875\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the answers obtained for this problem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+=+187.3125 ALIGN=MIDDLE  ALT=\"R+=+187.3125\"   DISABLED_style= \"max-width:90%; height:auto;\" > ohms and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5Bt%5D+=+645.1875 ALIGN=MIDDLE  ALT=\"R%5Bt%5D+=+645.1875\"   DISABLED_style= \"max-width:90%; height:auto;\" > ohms<BR>\n" );
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document.write( "Check the math above to make sure I didn't make some \"fat finger\" error on the calculator or<BR>\n" );
document.write( "in the process.<BR>\n" );
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document.write( "Hope this helps to clarify the general process a little, and I hope my initial assumption<BR>\n" );
document.write( "regarding the Gain formula was correct. <BR>\n" );
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