document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1000879>Question 1000879</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose you want to make a closed box with a square base. The material for the top and the bottom of the box costs $2 per sq in and the material for the sides costs $3 per sq in. What are the dimensions of the box of the greatest volume that can be constructed for $60? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618228 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=618228\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can solve this graphically or you can solve it using calculus.<BR>\n" );
document.write( "i don't know any other ways.<BR>\n" );
document.write( "the graphical solution is shown below:<BR>\n" );
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document.write( "that 2.236 turns out to be sqrt(5).<BR>\n" );
document.write( "the value of x is sqrt(5).<BR>\n" );
document.write( "the value of y is 7.454\r<BR>\n" );
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document.write( "x is the length of a side of the square base.<BR>\n" );
document.write( "y is the volume.\r<BR>\n" );
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document.write( "the formula for volume is derived as follows:\r<BR>\n" );
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document.write( "s = measure of one of the sides of the base.<BR>\n" );
document.write( "h = measure of the height.<BR>\n" );
document.write( "area of the base = s^2<BR>\n" );
document.write( "volume = s^2*h\r<BR>\n" );
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document.write( "surface area is derived as follows:<BR>\n" );
document.write( "sa = 2*s^2 + 4*h*s\r<BR>\n" );
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document.write( "cost of surface area is derived as follows:\r<BR>\n" );
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document.write( "cost of the base material is equal to 2 * the area of the base.<BR>\n" );
document.write( "cost of the side material is equal to 3 * the area of the sides between the 2 bases.\r<BR>\n" );
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document.write( "the total cost is there equal to 2 * area of the bases plus 3 * area of the side faces.\r<BR>\n" );
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document.write( "you get total cost = 2 * (2s^2) + 3*(4hs) which becomes:\r<BR>\n" );
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document.write( "total cost = 4s^2 + 12hs\r<BR>\n" );
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document.write( "since total cost is 60, you get:\r<BR>\n" );
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document.write( "60 = 4s^2 + 12hs\r<BR>\n" );
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document.write( "in this equation, you can solve for h as follows:\r<BR>\n" );
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document.write( "subtract 4s^2 from both sides to get:<BR>\n" );
document.write( "60 - 4s^2 = 12hs<BR>\n" );
document.write( "divide both sides by 12s to get:<BR>\n" );
document.write( "(60-4s^2) / 12s = h\r<BR>\n" );
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document.write( "the volume is equal to s^2*h<BR>\n" );
document.write( "replace h with (60-4s^2)/12s to get:<BR>\n" );
document.write( "volume = s^2 * (60-4s^2)/12s<BR>\n" );
document.write( "factor out an s from the numerator and denominator and you get:<BR>\n" );
document.write( "volume = s*(60-4s^2)/12\r<BR>\n" );
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document.write( "to graph this equation, make volume = y and make s = x.<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "y = x*(60-4x^2)/12<BR>\n" );
document.write( "this is the equation that was graphed above.\r<BR>\n" );
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document.write( "to solve this using calculus, get the derivative of the equation and set it equal to 0 and solve for x to find the maximum point on the graph.\r<BR>\n" );
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document.write( "the derivative turns out to be y' = 5-x^2<BR>\n" );
document.write( "set it equal to 0 and you get 0 = 5-x^2<BR>\n" );
document.write( "add x^2 to both sides to get x^2 = 5<BR>\n" );
document.write( "take square root of both sides to get x = +/- sqrt(5).<BR>\n" );
document.write( "it has to be sqrt(5) because negative values are not allowed.<BR>\n" );
document.write( "the derivative is telling you that you need to evalute your equation at x = sqrt(5) to find the maximum volume.<BR>\n" );
document.write( "that turns out to be same as what the graph is showing you.\r<BR>\n" );
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document.write( "the more detailed answer is:\r<BR>\n" );
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document.write( "x = s  = 2.236068<BR>\n" );
document.write( "y volume = 7.4535599\r<BR>\n" );
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document.write( "round these to x = 2.236 and y = 7.454\r<BR>\n" );
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document.write( "x is equal to s which is the width<BR>\n" );
document.write( "y is the volume\r<BR>\n" );
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document.write( "the volume is s^2*h<BR>\n" );
document.write( "you can use this formula to solve for h.<BR>\n" );
document.write( "from this formula, solve for h to get h = v/s^2.<BR>\n" );
document.write( "that becomes h = 7.4535599/2.236068 = 1.49071195<BR>\n" );
document.write( "round to 3 decimal place to get h = 1.491\r<BR>\n" );
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document.write( "the dimension of the box with the greatest volume that has a surface area that costs 60 dollars is therefore:\r<BR>\n" );
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document.write( "s = 2.236 <BR>\n" );
document.write( "h = 1.491\r<BR>\n" );
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document.write( "if you recall, we made s = x when we graphed it.<BR>\n" );
document.write( "when we found x, we automatically found s because they're equivalent to each other.\r<BR>\n" );
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document.write( "surface area = 2 * s^2 + 4hs\r<BR>\n" );
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document.write( "cost for surface area = 4s^2 + 12hs\r<BR>\n" );
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document.write( "when s = 2.236 and h = 1.491, we get cost for surface area = 59.999999999 which rounds to 60.\r<BR>\n" );
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document.write( "we got the maximum volume for a box that has a total cost for surface area of 60 dollars.\r<BR>\n" );
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document.write( "derivative was found using the following derivative calculator.\r<BR>\n" );
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document.write( "the calculator is very useful when you're not sure how to find the derivative.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.derivative-calculator.net/#\" target = \"_blank\">http://www.derivative-calculator.net/#</a>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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