document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150480>Question 1150480</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an right triangle prism the triangular base ABC is right angled at B,AB=5x cm and BC=12xcm.The sum of the lengths of all its edges is 18ocm.<BR>\n" );
document.write( "(a) Show that the volume,V cm^3, is given by V=1800x^2-600x^3<BR>\n" );
document.write( "(b)Find the value of x for which V has a maximum value </SPAN>\n" );
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document.write( "(a) Find an expression for the volume in terms of x.<br><BR>\n" );
document.write( "The two triangular bases have side lengths 5x, 12x, and 13x.<br><BR>\n" );
document.write( "The sum of the lengths of the sides of the two bases is 60x.<br><BR>\n" );
document.write( "The sum of the lengths of all the edges of the prism -- the sides of the bases, plus the three edges connecting the two bases -- is 180.<br><BR>\n" );
document.write( "Let h be the height of the prism; then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60x%2B3h+=+180 ALIGN=MIDDLE  ALT=\"60x%2B3h+=+180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3h+=+180-60x ALIGN=MIDDLE  ALT=\"3h+=+180-60x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+60-20x ALIGN=MIDDLE  ALT=\"h+=+60-20x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The volume is the area of the base, times the height:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%28%281%2F2%29%2812x%29%285x%29%29%2860-20x%29 ALIGN=MIDDLE  ALT=\"V+=+%28%281%2F2%29%2812x%29%285x%29%29%2860-20x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+30x%5E2%2860-20x%29+=+1800x%5E2-600x%5E3 ALIGN=MIDDLE  ALT=\"V+=+30x%5E2%2860-20x%29+=+1800x%5E2-600x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(b) Find the value of x that maximizes the volume.  (Find the value of x for which the derivative is zero.)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+=+3600x-1800x%5E2 ALIGN=MIDDLE  ALT=\"dV%2Fdx+=+3600x-1800x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3600x-1800x%5E2+=+0 ALIGN=MIDDLE  ALT=\"3600x-1800x%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800x%282-x%29+=+0 ALIGN=MIDDLE  ALT=\"1800x%282-x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2 ALIGN=MIDDLE  ALT=\"x+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note x=0 also makes the derivative zero but makes no sense in the actual problem.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );