document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204616>Question 1204616</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a cylinder with a surface area of 10<BR>\n" );
document.write( ", what is the maximum volume that it can have? Round your answer to the nearest 4 decimal places.\r<BR>\n" );
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document.write( "Recall that the volume of a cylinder is πr2h<BR>\n" );
document.write( " and the surface area is 2πrh+2πr2<BR>\n" );
document.write( " where r<BR>\n" );
document.write( " is the radius and h<BR>\n" );
document.write( " is the height.<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 #840973 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=840973\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "r = radius<BR>\n" );
document.write( "h = height\r<BR>\n" );
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document.write( "SA = surface area of the cylinder<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SA+=+2%2Api%2Ar%2Ah+%2B+2%2Api%2Ar%5E2 ALIGN=MIDDLE  ALT=\"SA+=+2%2Api%2Ar%2Ah+%2B+2%2Api%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+=+2%2Api%2Ar%2Ah+%2B+2%2Api%2Ar%5E2 ALIGN=MIDDLE  ALT=\"10+=+2%2Api%2Ar%2Ah+%2B+2%2Api%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given surface area of 10 square units. From here we isolate h.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+=+2%2A%28pi%2Ar%2Ah+%2B+pi%2Ar%5E2%29 ALIGN=MIDDLE  ALT=\"10+=+2%2A%28pi%2Ar%2Ah+%2B+pi%2Ar%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F2+=+pi%2Ar%2Ah+%2B+pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"10%2F2+=+pi%2Ar%2Ah+%2B+pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+=+pi%2Ar%2Ah+%2B+pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"5+=+pi%2Ar%2Ah+%2B+pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5-pi%2Ar%5E2+=+pi%2Ar%2Ah ALIGN=MIDDLE  ALT=\"5-pi%2Ar%5E2+=+pi%2Ar%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%285+-+pi%2Ar%5E2%29%2F%28pi%2Ar%29 ALIGN=MIDDLE  ALT=\"h+=+%285+-+pi%2Ar%5E2%29%2F%28pi%2Ar%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We'll use this so we can eliminate the variable h in the next equation below.\r<BR>\n" );
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document.write( "V = volume of the cylinder<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+pi%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"V+=+pi%2Ar%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+pi%2Ar%5E2%2A%28%285+-+pi%2Ar%5E2%29%2F%28pi%2Ar%29%29 ALIGN=MIDDLE  ALT=\"V+=+pi%2Ar%5E2%2A%28%285+-+pi%2Ar%5E2%29%2F%28pi%2Ar%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the equation we solved previously\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+r%2A%285+-+pi%2Ar%5E2%29 ALIGN=MIDDLE  ALT=\"V+=+r%2A%285+-+pi%2Ar%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+5r+-+pi%2Ar%5E3 ALIGN=MIDDLE  ALT=\"V+=+5r+-+pi%2Ar%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > We end up with the volume in terms of one single variable.\r<BR>\n" );
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document.write( "Let's consider the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+5x+-+pi%2Ax%5E3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+5x+-+pi%2Ax%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = r = radius<BR>\n" );
document.write( "f(x) = V = volume\r<BR>\n" );
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document.write( "Domain: x > 0<BR>\n" );
document.write( "Range: f(x) > 0<BR>\n" );
document.write( "We focus on the upper right quadrant (aka Q1)\r<BR>\n" );
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document.write( "Use a graphing calculator, or derivatives (if you are in a calculus class), to find the highest point in Q1 occurs at the approximate location of (0.728366, 2.427885)<BR>\n" );
document.write( "GeoGebra and Desmos are two graphing options that I recommend.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-100%2C5x+-+pi%2Ax%5E3%29%2C%0D%0Acircle%280.728366%2C2.427885%2C0.05%29%2Ccircle%280.728366%2C2.427885%2C0.07%29%2Ccircle%280.728366%2C2.427885%2C0.09%29%2Ccircle%280.728366%2C2.427885%2C0.11%29%2C%0D%0Alocate%280.728366%2B0.2%2C+2.427885%2B0.2%2C%22%280.728366%2C+2.427885%29%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Here is the link to the interactive Desmos graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/pzk6yfzxpd\">https://www.desmos.com/calculator/pzk6yfzxpd</a><BR>\n" );
document.write( "Click on the highest point to have its coordinates show up. You may have to click twice.\r<BR>\n" );
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document.write( "Therefore a radius of approximately 0.728366 units leads to the max cylinder volume of approximately 2.427885 cubic units. This applies only when the surface area is 10 square units.<BR>\n" );
document.write( "A real world application is that you have 10 square units of material, and the goal is to get the most storage space out of the cylinder.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>2.4279 cubic units</font><BR>\n" );
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