document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181985>Question 1181985</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a baking pan has an rectangular base 12 in by 8 in, the sides and ends of the pan slope outward, so that the upper edges measure respectively 13 1/2 in by 9 in. if the depth of the pan is 2 in, find the amount of cake batter requires to fill the pan to 1/2 its depth </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812519 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=812519\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
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document.write( "My visualization of the problem was faulty.  The \"pyramids\" in the corners of the pan have rectangular bases, not triangular.<br><BR>\n" );
document.write( "Furthermore, the problem is easier than I made it, because the 8x12 dimensions of the bottom of the pan and the 9x13.5 dimensions of the top make those two rectangles similar; and that means the problem can be solved using the idea of a truncated pyramid.<br><BR>\n" );
document.write( "Thanks to tutor @ikleyn for providing a solution by that method.<br><BR>\n" );
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document.write( "UGLY problem....  But I finally decided to buckle down and tackle it.<br><BR>\n" );
document.write( "For calculating the volume of the pan, it can be divided up into several pieces.  For the purpose of this problem, we will express the volume of each piece as a function of h, the height/depth.<br><BR>\n" );
document.write( "(1) The largest piece is a rectangular prism, dimensions (12)x(8)x(h).<br><BR>\n" );
document.write( "Volume of rectangular prism: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%29%288%29%28h%29+=+96h ALIGN=MIDDLE  ALT=\"%2812%29%288%29%28h%29+=+96h\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "(2) Next there are triangular prisms along each side of the pan.<br><BR>\n" );
document.write( "The base of the pan is 8x12 inches.  With the sloping sides of the pan, the dimensions of the top edge of the pan are 9x13.5 inches.<br><BR>\n" );
document.write( "That means along the long side of the pan the lengths of the top and bottom edges differ by (9-8)/2 = 0.5 = 1/2 inches; and along the short side of the pan the top and bottom edges differ by (13.5-12)/2 = 0.75 = 3/4 inches.<br><BR>\n" );
document.write( "But those are the \"overlaps\" for the full pan, with a depth of 2 inches.  For solving this problem we will need to write those overlaps as functions of the depth h.  So the overlap along the long edge, 1/2 inch when the depth is 2 inches, is (1/4)h.  And the overlap along the short edge, 3/4 inch when the depth is 2 inches, is (3/8)h.<br><BR>\n" );
document.write( "So we have two triangular prisms along the two long edges; the legs of the triangular base are h and (1/4)h, and the length is 12.<br><BR>\n" );
document.write( "Volume of those two triangular prisms: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%29%28%281%2F2%29%28h%29%28%281%2F4%29h%29%29%2812%29+=+3h%5E2 ALIGN=MIDDLE  ALT=\"%282%29%28%281%2F2%29%28h%29%28%281%2F4%29h%29%29%2812%29+=+3h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Similarly we have two triangular prisms along the two short edges; the legs of the triangular base are h and (3/8)h, and the length is 8.<br><BR>\n" );
document.write( "Volume of those two triangular prisms: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%29%28%281%2F2%29%28h%29%28%283%2F8%29h%29%29%288%29+=+3h%5E2 ALIGN=MIDDLE  ALT=\"%282%29%28%281%2F2%29%28h%29%28%283%2F8%29h%29%29%288%29+=+3h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(3) Lastly, in each of the four corners of the pan we have a pyramid with height h and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28bases%29 ALIGN=MIDDLE  ALT=\"cross%28bases%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%281%2F4%29h%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F4%29h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28and%29 ALIGN=MIDDLE  ALT=\"cross%28and%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%283%2F8%29h%29 ALIGN=MIDDLE  ALT=\"cross%28%283%2F8%29h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > rectangular bases (1/4)h by (3/8)h.<br><BR>\n" );
document.write( "Volume of the four pyramids: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%284%29%28%281%2F3%29%28%281%2F2%29%28%281%2F4%29h%29%28%283%2F8%29h%29%29%29%28h%29+=+%281%2F16%29h%5E3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%28%281%2F3%29%28%28%281%2F4%29h%29%28%283%2F8%29h%29%29%29%28h%29+=+%281%2F8%29h%5E3 ALIGN=MIDDLE  ALT=\"%284%29%28%281%2F3%29%28%28%281%2F4%29h%29%28%283%2F8%29h%29%29%29%28h%29+=+%281%2F8%29h%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the volume of whatever is in the pan as a function of the depth is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28h%29+=+96h%2B6h%5E2%2B%281%2F8%29h%5E3 ALIGN=MIDDLE  ALT=\"V%28h%29+=+96h%2B6h%5E2%2B%281%2F8%29h%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We are to find the volume of the batter when the pan is filled to half its depth of 2 inches; so we need to evaluate this volume function for h=1.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%281%29+=+96%2B6%2B1%2F8 ALIGN=MIDDLE  ALT=\"V%281%29+=+96%2B6%2B1%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 102.125 cubic inches.<br><BR>\n" );
document.write( "Note the volume of the full pan is V(2) = 192+24+1/2 = 216.5 cubic inches; the amount of batter to fill the pan to half its depth should be a bit less than half the full volume, so our answer of about 102 cubic inches is reasonable.<br><BR>\n" );
document.write( "ANSWER: 102.125 cubic inches of batter will fill the pan to half its depth.<br><BR>\n" );
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