document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=148659>Question 148659</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321356918/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Please helpo solve! <BR>\n" );
document.write( "i)	Problem: A sheet metal worker is planning to make an open-top box by cutting equal squares (x-in. by x-in.) from the corners of a 10-in. by 14-in. piece of copper. A second box is to be made in the same manner from an 8-in. by 10-in. piece of aluminum, but its height is to be one-half that of the first box.<BR>\n" );
document.write( "(1)Find a polynomial function for the volume of each box.<BR>\n" );
document.write( "(2)Find the values of x for which the copper box is 72 cubic in. larger than the aluminum box.<BR>\n" );
document.write( "(3)Write the difference between the two volumes (d) as a function of x.<BR>\n" );
document.write( "(4)Find d for x=1.5.<BR>\n" );
document.write( "(5)For what value of x is the difference between the two volumes the largest?  <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #109115 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=109115\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i) Problem: A sheet metal worker is planning to make an open-top box by cutting equal squares (x-in. by x-in.) from the corners of a 10-in. by 14-in. piece of copper. A second box is to be made in the same manner from an 8-in. by 10-in. piece of aluminum, but its height is to be one-half that of the first box.<BR>\n" );
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document.write( "(1)Find a polynomial function for the volume of each box.<BR>\n" );
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document.write( "1st box dimensions: (10-2x) by (14-2x) by x<BR>\n" );
document.write( "Vol = x*(10-2x)*(14-2x)<BR>\n" );
document.write( "FOIL<BR>\n" );
document.write( "f(x) = x(140 - 48x + 4x^2)<BR>\n" );
document.write( "f(x) = 4x^3 - 48x^2 + 140x<BR>\n" );
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document.write( "2nd box cut out will be .5x in. by .5x in.,(height 1/2 the 1st box), therefore:<BR>\n" );
document.write( " the dimensions will be:<BR>\n" );
document.write( "(10-x) by (8-x) by .5x<BR>\n" );
document.write( "Vol = .5x*(10-x)*(8-x)<BR>\n" );
document.write( "FOIL<BR>\n" );
document.write( "f(x) = .5x(80 - 18x + x^2)<BR>\n" );
document.write( "f(x) = .5x^3 - 9x^2 + 40x<BR>\n" );
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document.write( ": <BR>\n" );
document.write( "(2)Find the values of x for which the copper box is 72 cubic in. larger than the aluminum box.<BR>\n" );
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document.write( "Big box vol - small box vol = 72 cu in<BR>\n" );
document.write( "(4x^3 - 48x^2 + 140x) - (.5x^3 - 9x^2 + 40x) = 72<BR>\n" );
document.write( "Remove brackets<BR>\n" );
document.write( "4x^3 - 48x^2 + 140x - .5x^3 + 9x^2 - 40x = 72<BR>\n" );
document.write( "Group like terms<BR>\n" );
document.write( "4x^3 - .5x^3 - 48x^2 + 9x^2 + 140x - 40x - 72 = 0<BR>\n" );
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document.write( "3.5x^3 - 39x^2 + 100x - 72 = 0<BR>\n" );
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document.write( "Solve this by graphing y = 3.5x^3 - 39x^2 + 100x - 72<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-4%2C+10%2C+-10%2C+10%2C+3.5x%5E3-39x%5E2%2B100x-72%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-4%2C+10%2C+-10%2C+10%2C+3.5x%5E3-39x%5E2%2B100x-72%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The integer solution x = 2; & x ~ 1.3; the third solution x ~ 7.8 isn't reasonable <BR>\n" );
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document.write( "(3)Write the difference between the two volumes (d) as a function of x.<BR>\n" );
document.write( "d(x) = (4x^3 - 48x^2 + 140x) - (.5x^3 - 9x^2 + 40x)<BR>\n" );
document.write( "which is<BR>\n" );
document.write( "d(x) = 3.5x^3 - 39x^2 + 100x<BR>\n" );
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document.write( "(4)Find d for x=1.5<BR>\n" );
document.write( "d(x) = 3.5(1.5^3) - 39(1.5^2) + 100(1.5)<BR>\n" );
document.write( "d(x) = 3.5(3.375) - 39(2.25) + 150<BR>\n" );
document.write( "d(x) = 11.8125 - 87.75 + 150<BR>\n" );
document.write( "d(x) = 74.0625<BR>\n" );
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document.write( "(5)For what value of x is the difference between the two volumes the largest? <BR>\n" );
document.write( "Graph d(x) = 3.5x^3 - 39x^2 + 100x: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-4%2C+6%2C+-20%2C+100%2C+3.5x%5E3-39x%5E2%2B100x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-4%2C+6%2C+-20%2C+100%2C+3.5x%5E3-39x%5E2%2B100x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Greatest difference appears to be at x = 1.7 which is a difference of about 74.5 </SPAN>\n" );
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