document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114309>Question 114309</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.  A piece of cardboard measuring 10 inches by 15 inches is to be made into a box with an open top by cutting equal-sized squares from each corner and folding up the sides.  Let x represent the length of a side of each such square in inches.\r<BR>\n" );
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document.write( "a)  Draw a diagram\r<BR>\n" );
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document.write( "b)  What are the restrictions on x?\r<BR>\n" );
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document.write( "c)  Determine a function V that gives the volume of the box.\r<BR>\n" );
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document.write( "d)  Draw a graph the function and find the value of x that produces the maximum volume.  What is the maximum volume of the box?\r<BR>\n" );
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document.write( "e)  When will the volume of the box be greater than 80 cu inches?\r<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 #83308 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=83308\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A piece of cardboard measuring 10 inches by 15 inches is to be made into a box with an open top by cutting equal-sized squares from each corner and folding up the sides. Let x represent the length of a side of each such square in inches.<BR>\n" );
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document.write( "a) Draw a diagram<BR>\n" );
document.write( "Draw a rectangle about 10 by 15 cm. Label it 10 by 15 inches. Draw a small square at each corner. Label it's side as x<BR>\n" );
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document.write( "b) What are the restrictions on x?<BR>\n" );
document.write( "From the diagram you can see the box dimensions will be (10-2x) by (15-2x)<BR>\n" );
document.write( "Therefore you know the value of x has to be less than 5. ie (10-2(5)) = 0<BR>\n" );
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document.write( "c) Determine a function V that gives the volume of the box.<BR>\n" );
document.write( "The length = (15-2x)<BR>\n" );
document.write( "The width = (10-2x)<BR>\n" );
document.write( "The height = x<BR>\n" );
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document.write( "V = (15-2x)*(10-2x) * x<BR>\n" );
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document.write( "V = (150 - 30x - 20x + 4x^2) * x<BR>\n" );
document.write( "Which is:<BR>\n" );
document.write( "V = x(4x^2 - 50x + 150)<BR>\n" );
document.write( "V = 4x^3 - 50x^2 + 150x; is the function of the volume<BR>\n" );
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document.write( "d) Draw a graph the function and find the value of x that produces the maximum <BR>\n" );
document.write( "volume.<BR>\n" );
document.write( "Plot your graph x = .5 to x = +5, every .5 inches <BR>\n" );
document.write( " x | y<BR>\n" );
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document.write( ".5 | 63 <BR>\n" );
document.write( "1.0|104<BR>\n" );
document.write( "1.5|126<BR>\n" );
document.write( "2.0|132<BR>\n" );
document.write( "2.5|125<BR>\n" );
document.write( "3.0|108<BR>\n" );
document.write( "3.5|84<BR>\n" );
document.write( "4.0|56<BR>\n" );
document.write( "4.5|27<BR>\n" );
document.write( "5.0| 0 <BR>\n" );
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document.write( "Your graph should look like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-2%2C+5%2C+-20%2C+140%2C+4x%5E3-50x%5E2%2B150x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-2%2C+5%2C+-20%2C+140%2C+4x%5E3-50x%5E2%2B150x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "What is the maximum volume of the box?<BR>\n" );
document.write( "Looking at the graph we can see max vol occurs when x = 2, vol = 132 cu in<BR>\n" );
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document.write( "Substitute 2 for x in the original equation to confirm this:<BR>\n" );
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document.write( "e) When will the volume of the box be greater than 80 cu inches?<BR>\n" );
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document.write( "Looking at the graph (and the table) we can say between 1 and 3.5 inches the vol > 80<BR>\n" );
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document.write( "You can confirm this also by substituting these values in the equation<BR>\n" );
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document.write( "Any questions about this? </SPAN>\n" );
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