document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=181288>Question 181288</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321233832/algebrahomeworkh>Fourth Edition Elementary and Intermediate Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> 1.	Steve traveled 150 miles at a certain speed. Had he gone 20mph faster, the trip would have taken 2 hours less. Find the speed of his vehicle.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #135929 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=135929\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Consider the basic formula that relates distance, rate (speed), and time:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math d = rt\">\r<BR>\n" );
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document.write( "and its variations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math r = \frac {d}{t}\">\r<BR>\n" );
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document.write( "and \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math t = \frac {d}{r}\">\r<BR>\n" );
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document.write( "It will be convenient to use the last one because that will ultimately allow us to solve for the desired quantity, <i>r</i>, directly.\r<BR>\n" );
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document.write( "The actual trip can be described thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math t = \frac {150}{r}\">\r<BR>\n" );
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document.write( "And the 'what if' (20 mph faster, 2 hours less) trip can be described:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math t - 2 = \frac {150}{r + 20} \ \ \Rightarrow \ \ t = \left(\frac{150}{r + 20}\right) + 2 \ \ \Rightarrow \ \ t = \left(\frac{150}{r + 20}\right) + \left(\frac {2(r + 20)}{r + 20}\right) \ \ \Rightarrow \ \ t = \frac {2r + 190}{r + 20} \">\r<BR>\n" );
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document.write( "Now we have expressed the variable <i>t</i> in two different ways in terms of <i>r</i>, the quantity we seek.  So set these two expressions in <i>r</i> equal to one another.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math \frac{150}{r} = \frac {2r + 190}{r + 20} \">\r<BR>\n" );
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document.write( "This is a simple proportion that can be solved by first cross-multiplying and simplifying:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math 150(r + 20) = 2r^2 + 190r \ \ \Rightarrow \ \ -2r^2 -40r + 3000 = 0 \ \ \Rightarrow \ \ r^2 + 20r - 1500 = 0\">\r<BR>\n" );
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document.write( "Leaving us with a factorable quadratic:\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -30 + 50 = 20\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -30 \times 50 = -1500\">, we can say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math r^2 + 20r - 1500 = 0 \ \ \Rightarrow \ \ (r - 30)(r + 50) = 0\">\r<BR>\n" );
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document.write( "Hence,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math  r = 30\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math  r = -50\">\r<BR>\n" );
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document.write( "But -50 for a speed is absurd and is therefore an extraneous root introduced by the action of squaring the variable during the solution process.  Exclude -50.\r<BR>\n" );
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document.write( "That leaves \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math  r = 30\"> as the answer.\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math  r = 30 \ \ \Rightarrow \ \ t = \frac {150}{30} = 5\">\r<BR>\n" );
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document.write( "If <i>r</i> were 20 mph faster, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r = 30 + 20 = 50 \"> and if <i>t</i> were 2 hours less, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE t = 5 - 2 = 3\">.  If the solution is correct, then a trip lasting 3 hours at 50 miles per hour should cover the same 150 mile distance.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text {          }\math d = rt = 50 \times 3 = 150\">\r<BR>\n" );
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document.write( "Answer checks.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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