document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161965>Question 1161965</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In this investigation, you will be estimating the values of and to the nearest hundredths place. \r<BR>\n" );
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document.write( "(this numbers I need the square root) (√40  √75  √90 √245 √450 √640 √675 )\r<BR>\n" );
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document.write( "    1. Describe clearly what strategy you plan to use to estimate these values.\r<BR>\n" );
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document.write( "Reflect on the calculations you made. What general methods or short cuts did you find to make the calculations easier as you did each calculation?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #785619 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=785619\">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( "I'll do one of these; you can figure out the rest.\r<BR>\n" );
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document.write( "Consider <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{90}\">.  What is the largest perfect square that is less than 90?  And what is the smallest perfect square that is larger than 90?  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 9\ \times\ 9\ =\ 81\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 10\ \times\ 10\ =\ 100\">, respectively.  So, since:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 81\ \leq\ 90\ \leq\ 100\">\r<BR>\n" );
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document.write( "we can say\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9\ \leq\ \sqrt{90}\ \leq\ 10\">\r<BR>\n" );
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document.write( "Since 90 is pretty close to the middle of the 81 to 100 interval, 9.5 is a pretty good initial guess for the square root of 90.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9.5^2\ =\ 90.25\">\r<BR>\n" );
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document.write( "A little bit too large, so let's try 9.4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9.4^2\ =\ 88.36\">\r<BR>\n" );
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document.write( "A lot too small compared to our first guess, but now we know that.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9.4\ \leq\ \sqrt{90}\ \leq\ 9.5\">\r<BR>\n" );
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document.write( "Since 9.5 squared is significantly closer to 90 than 9.4 squared, the next guess should be closer to 9.5 than 9.4.  Let's try the value 9.47.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9.47^2\ =\ 89.68\">\r<BR>\n" );
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document.write( "Close, but still a little small, so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9.49^2\ =\ 89.96\">\r<BR>\n" );
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document.write( "We are still lower, but closer to the true value than our initial guess of 9.5, but now we know that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9.49\ \leq\ \sqrt{90}\ \leq\ 9.50\">\r<BR>\n" );
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document.write( "Since we have seen that 9.49 is closer than 9.50 and you only have to get to the nearest hundredths place, 9.49 is your answer.\r<BR>\n" );
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document.write( "Checking the work we see that the calculator value is:\r<BR>\n" );
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document.write( "9.4868329805051379959966806332982 which rounds to 9.49.  By the way, remember that the calculator value is still an approximation, albeit a much better one than the above process.  But the square root of anything that is not a perfect square is an irrational number -- the decimal representation goes on forever and never repeats.  The only way to represent the square root of 90 exactly is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{90}\">\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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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