document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=61442>Question 61442</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0534419860/algebrahomeworkh>Calculus for the Managerial, Life, and Social Sciences</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular box is to have a square base and a volume of 20 feet cubed.  If the material for the base costs $.30/square foot, the material for the sides costs $.10/square foot, and the material for the top costs $.20/square foot, determine the dimensions of the box that can be constructed at minimun cost. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #42311 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=42311\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A rectangular box is to have a square base and a volume of 20 feet cubed. If the material for the base costs $.30/square foot, the material for the sides costs $.10/square foot, and the material for the top costs $.20/square foot, determine the dimensions of the box that can be constructed at minimum cost<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Let y = the cost of the box<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Let x = one side of the square base<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Let h = height of the box<BR>\n" );
document.write( ":<BR>\n" );
document.write( "We know that the box has to be 20 cu ft:<BR>\n" );
document.write( "x^2*h = 20<BR>\n" );
document.write( "h = (20/x^2)<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Costs:<BR>\n" );
document.write( "Base = .30(x^2)<BR>\n" );
document.write( "Top = .20(x^2)<BR>\n" );
document.write( "1 side = .10(x*(20/x^2)) = .10(20/x)<BR>\n" );
document.write( "4 sides = .40(20/x) = 8/x <BR>\n" );
document.write( ":<BR>\n" );
document.write( "Total cost:<BR>\n" );
document.write( "y =  Base + Top + 4 sides <BR>\n" );
document.write( "y =.30(x^2) + .20(x^2) + 8/x<BR>\n" );
document.write( "y = .5x^2 + 8/x<BR>\n" );
document.write( "Graph it:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-6%2C+6%2C+-6%2C+12%2C+.5x%5E2+%2B+%288%2Fx%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-6%2C+6%2C+-6%2C+12%2C+.5x%5E2+%2B+%288%2Fx%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ":<BR>\n" );
document.write( "Looks like min cost (y) occurs when x = 2 ft, looks like a cost(y) of $6.00<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Then the height = 20/2^2 = 5 ft<BR>\n" );
document.write( ":<BR>\n" );
document.write( "What is the cost then?<BR>\n" );
document.write( "top .30(2^2) = 1.20<BR>\n" );
document.write( "bot .20(2^2) = 0.80<BR>\n" );
document.write( "hth .10(2*5) = 1.00 * 4 sides = $4.00<BR>\n" );
document.write( "Total cost = $6.00 like the graph indicates.<BR>\n" );
document.write( " <BR>\n" );
document.write( ":<BR>\n" );
document.write( "Check: 2 * 2 * 5 = 20 cu ft as required\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );