document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099840>Question 1099840</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am in College Algebra and need some major help. A cylinder shaped can needs to be constructed to hold 600 cubic centimeters of soup. The material for the sides of the can costs 0.04 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.05 cents per square centimeter. Find the dimensions for the can that will minimize production cost. <BR>\n" );
document.write( "Helpful information:<BR>\n" );
document.write( "h : height of can, r : radius of can <BR>\n" );
document.write( "Volume of a cylinder: V=&#960;r2h <BR>\n" );
document.write( "Area of the sides: A=2&#960;rh<BR>\n" );
document.write( "Area of the top/bottom: A=&#960;r<BR>\n" );
document.write( "To minimize the cost of the can:<BR>\n" );
document.write( "The radius should be________.<BR>\n" );
document.write( "The minimum cost should be________cents.<BR>\n" );
document.write( "The height should be________. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #714450 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=714450\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The surface area of the can is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+2%2Api%2Arh+%2B+2%2Api%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A+=+2%2Api%2Arh+%2B+2%2Api%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <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;\" >, we can solve for h in terms of r:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+V%2F%28pi%2Ar%5E2%29 ALIGN=MIDDLE  ALT=\"h+=+V%2F%28pi%2Ar%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, rather than simply minimizing the surface area, we need to minimize the cost<BR>\n" );
document.write( "function, since the sides and top/bottom have a different cost per unit area.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28r%2Ch%29+=+0.04%2A2%2Api%2Arh+%2B+0.05%2A2%2Api%2Ar%5E2 ALIGN=MIDDLE  ALT=\"C%28r%2Ch%29+=+0.04%2A2%2Api%2Arh+%2B+0.05%2A2%2Api%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28r%2Ch%29+=+0.08%2Api%2Arh+%2B+0.1%2Api%2Ar%5E2 ALIGN=MIDDLE  ALT=\"C%28r%2Ch%29+=+0.08%2Api%2Arh+%2B+0.1%2Api%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the value of h above, we have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28r%29+=+0.08V%2Fr+%2B+0.1%2Api%2Ar%5E2 ALIGN=MIDDLE  ALT=\"C%28r%29+=+0.08V%2Fr+%2B+0.1%2Api%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The cost function will be minimized where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2Fdr+=+0 ALIGN=MIDDLE  ALT=\"dC%2Fdr+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2Fdr+=+0+=+-0.08V%2Fr%5E2+%2B+0.2%2Api%2Ar ALIGN=MIDDLE  ALT=\"dC%2Fdr+=+0+=+-0.08V%2Fr%5E2+%2B+0.2%2Api%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving for r, we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%280.02V%2F0.05%2Fpi%29%5E%281%2F3%29+=+%282V%2F5%2Fpi%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"r+=+%280.02V%2F0.05%2Fpi%29%5E%281%2F3%29+=+%282V%2F5%2Fpi%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+V%2Fpi%2F%28%282V%2F5%2Fpi%29%5E%282%2F3%29%29 ALIGN=MIDDLE  ALT=\"h+=+V%2Fpi%2F%28%282V%2F5%2Fpi%29%5E%282%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the values, we get:<BR>\n" );
document.write( "r = 4.243 cm<BR>\n" );
document.write( "h = 10.608 cm<BR>\n" );
document.write( "C = 16.97 cents<BR>\n" );
document.write( "etc...<BR>\n" );
document.write( "The derivative of the cost function with respect to r is shown below.<BR>\n" );
document.write( "The function is zero at r = 4.243 cm <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C300%2C-4%2C12%2C-20%2C20%2C-48%2Fx%5E2+%2B+0.314159%2A2%2Ax%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C300%2C-4%2C12%2C-20%2C20%2C-48%2Fx%5E2+%2B+0.314159%2A2%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );