document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=366108>Question 366108</A>: <I><SPAN STYLE=\"word-break: break-all;\"> John is making juice. For this particular juice, he's filling a bottle in the shape of a pentagonal pyramid: the base is a regular pentagon, 15cm to a side, and the height of the \"pyramid\" is 32cm. Assume these measurements are for the interior of the bottle.\r<BR>\n" );
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document.write( "Assume the flow of juice into the bottle starts at zero and increases at a rate of 1 millilitre per second per second. How many seconds will it take to completely fill the bottle? Please round to the nearest whole number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #260889 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=260889\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r'\"> represent the rate of change of the fill rate; in this case <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\ ml/sec^2\">.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\"> be the instantaneous fill rate, and let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V\"> be the instantaneous volume filled.  And finally, let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> represent the elapsed time in seconds since I started to fill the bottle.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ \int\,1\ \text{ml/sec^2}\ dt\ =\ t\ \text{ml/sec}\ +\ C\">\r<BR>\n" );
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document.write( "But the constant of integration in this case is the initial flow rate, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ t\ +\ r_o\ \text{ml/sec}\">\r<BR>\n" );
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document.write( "Then the instantaneous volume is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V\ =\ \int\ r\ dt\ =\ \int\ t\ +\ r_o\ \text{ml/sec}\ dt\ =\ \frac{t^2}{2}\ +\ r_ot\ +\ V_o\ \text{ml}\">.\r<BR>\n" );
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document.write( "Since both the initial volume, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V_o\">, and the initial fill rate, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r_o\">, are both zero, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V_p\"> is the volume of the pyramid bottle, then we only need solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{t^2}{2}\ =\ V_p\">\r<BR>\n" );
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document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\">, namely:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \sqrt{\frac{V_p}{2}}\ \text{sec}\">\r<BR>\n" );
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document.write( "since we can assume for the precision required for this problem that cubic centimeters are equivalent to milliliters.\r<BR>\n" );
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document.write( "The volume of any pyramid is given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{3}Bh\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h\"> is the height, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large B\"> is the area of the base.\r<BR>\n" );
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document.write( "To find the area of a regular polygon knowing the measure of a side, use:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A\ =\ \frac{1}{4}ns^2\cot\left(\frac{180^\circ}{n}\right)\">\r<BR>\n" );
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document.write( "Which, for your regular pentagon reduces to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A_{pentagon}\ =\ \frac{5}{4}s^2\cot\left(36^\circ\right)\">\r<BR>\n" );
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document.write( "You should be able to handle it from here.\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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