document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=728575>Question 728575</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help solving this word problem:<BR>\n" );
document.write( "A box with no lid is to be constructed out of a piece of cardboard 17'' by 22'' by cutting out squares x'' per side in each corner and then folding up the sides. If the volume of the resulting box is to be 152.5 cubic inches, find the sizes of the cut-outs. Round your answear(s) to nearest hundredth. What is your domain? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #445627 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=445627\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-2%2C24%2C-1.5%2C17.5%2C%0D%0Ared%28rectangle%280%2C0%2C3%2C3%29%29%2Cred%28rectangle%2819%2C0%2C22%2C3%29%29%2C%0D%0Ared%28rectangle%280%2C14%2C3%2C17%29%29%2Cred%28rectangle%2819%2C14%2C22%2C17%29%29%2C%0D%0Ablue%28rectangle%283%2C3%2C19%2C14%29%29%2Crectangle%280%2C0%2C22%2C17%29%2C%0D%0Alocate%2810%2C0%2C22%29%2Clocate%2822.2%2C9%2C17%29%2C%0D%0Alocate%281.3%2C14%2Cx%29%2Clocate%283.2%2C16%2Cx%29%2C%0D%0Alocate%289%2C14%2C22-2x%29%2Clocate%283.2%2C9%2C17-2x%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Cut along red lines; fold along blue lines.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=The+blue+base+has+a+surface+area+of%0D%0A%7B%7B%7B%2822-2x%29%2817-2x%29 ALIGN=MIDDLE  ALT=\"The+blue+base+has+a+surface+area+of%0D%0A%7B%7B%7B%2822-2x%29%2817-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > square inches.<BR>\n" );
document.write( "The box has a height of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches.<BR>\n" );
document.write( "The volume of the box is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=x%2822-2x%29%2817-2x%29 ALIGN=MIDDLE  ALT=\"V=x%2822-2x%29%2817-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >cubic inches.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The expression above gives you the volume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the measure of the side of the squares cut off imposes some restrictions:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17-2x%3E0 ALIGN=MIDDLE  ALT=\"17-2x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17%3E2x ALIGN=MIDDLE  ALT=\"17%3E2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17%2F2%3Ex ALIGN=MIDDLE  ALT=\"17%2F2%3Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C8.5 ALIGN=MIDDLE  ALT=\"x%3C8.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C8%261%2F2 ALIGN=MIDDLE  ALT=\"x%3C8%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the variable <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > must comply with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C8.5 ALIGN=MIDDLE  ALT=\"0%3Cx%3C8.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is the domain of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=152.5 ALIGN=MIDDLE  ALT=\"V=152.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > cubic inches, so we must solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2822-2x%29%2817-2x%29=152.5 ALIGN=MIDDLE  ALT=\"x%2822-2x%29%2817-2x%29=152.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and find one or more solutions with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C8.5 ALIGN=MIDDLE  ALT=\"0%3Cx%3C8.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is a cubic (degree 3) equation, and may be difficult to solve.<BR>\n" );
document.write( "There could be a rational solution somewhere that could be used to reduce the equation to a quadratic equation.<BR>\n" );
document.write( "Otherwise, if the solutions turn out to be irrational, it is a question of finding approximate solutions by painful calculations. The request to round answers to the nearest hundredth suggests that is the case.<BR>\n" );
document.write( "A graphing calculator, or some computer software could help with those calculations. The question about the domain suggest that using a graphing calculator is expected. If you have been using a graphing calculator in class, that must be what is expected.<BR>\n" );
document.write( "The polynomial function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=x%2822-2x%29%2817-2x%29 ALIGN=MIDDLE  ALT=\"V=x%2822-2x%29%2817-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be zero at x=0 and x=8.5, and we know that it is positive in between.<BR>\n" );
document.write( "Logic tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=x%2822-2x%29%2817-2x%29 ALIGN=MIDDLE  ALT=\"V=x%2822-2x%29%2817-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a maximum in the domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C8.5 ALIGN=MIDDLE  ALT=\"0%3Cx%3C8.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=1%2A20%2A15=300%3E152.5 ALIGN=MIDDLE  ALT=\"V=1%2A20%2A15=300%3E152.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so 152.5 is not the maximum.<BR>\n" );
document.write( "We should expect two solutions, one on each side of the maximum.<BR>\n" );
document.write( "I find that there is a solution between 0.445 and 0.450, which would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280.45%29 ALIGN=MIDDLE  ALT=\"highlight%280.45%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > rounded to the nearest hundredth.<BR>\n" );
document.write( "There is another solution between 7.115 ans 7.12 which would round to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%287.12%29 ALIGN=MIDDLE  ALT=\"highlight%287.12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );