document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198241>Question 1198241</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A square pyramid whose lateral edge is 3 ¼ times its base edge can hold 250 cubic inches of sand when level full. How many square inches are there in its lateral surface? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848328 by </B><A HREF=/tutors/aboutme.mpl?userid=onyulee><B>onyulee(41)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=onyulee><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848328\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define Variables**\r<BR>\n" );
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document.write( "* Let 's' be the side length of the square base.<BR>\n" );
document.write( "* Let 'l' be the length of the lateral edge (l = 3.25s).<BR>\n" );
document.write( "* Let 'h' be the height of the pyramid.\r<BR>\n" );
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document.write( "**2. Find the Volume of the Pyramid**\r<BR>\n" );
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document.write( "The volume of a square pyramid is given by:\r<BR>\n" );
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document.write( "* Volume = (1/3) * Base Area * Height <BR>\n" );
document.write( "* 250 = (1/3) * s^2 * h\r<BR>\n" );
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document.write( "**3. Find the Height of the Pyramid**\r<BR>\n" );
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document.write( "* We know that the lateral edge, base edge, and height form a right triangle. <BR>\n" );
document.write( "* Using the Pythagorean Theorem:<BR>\n" );
document.write( "    * h^2 + (s/2)^2 = l^2 <BR>\n" );
document.write( "    * h^2 + (s^2)/4 = (3.25s)^2<BR>\n" );
document.write( "    * h^2 = (3.25s)^2 - (s^2)/4<BR>\n" );
document.write( "    * h^2 = 10.5625s^2 - 0.25s^2<BR>\n" );
document.write( "    * h^2 = 10.3125s^2<BR>\n" );
document.write( "    * h = √(10.3125s^2) <BR>\n" );
document.write( "    * h = s√10.3125\r<BR>\n" );
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document.write( "**4. Substitute 'h' in the Volume Equation**\r<BR>\n" );
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document.write( "* 250 = (1/3) * s^2 * (s√10.3125)<BR>\n" );
document.write( "* 250 = (√10.3125/3) * s^3<BR>\n" );
document.write( "* s^3 = 250 / (√10.3125/3)<BR>\n" );
document.write( "* s^3 ≈ 22.81<BR>\n" );
document.write( "* s ≈ 2.84 inches\r<BR>\n" );
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document.write( "**5. Calculate the Lateral Edge Length**\r<BR>\n" );
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document.write( "* l = 3.25s <BR>\n" );
document.write( "* l = 3.25 * 2.84 <BR>\n" );
document.write( "* l ≈ 9.21 inches\r<BR>\n" );
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document.write( "**6. Calculate the Slant Height**\r<BR>\n" );
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document.write( "* Let 'L' be the slant height.<BR>\n" );
document.write( "* L^2 = h^2 + (s/2)^2<BR>\n" );
document.write( "* L^2 = (s√10.3125)^2 + (s/2)^2<BR>\n" );
document.write( "* L^2 = 10.3125s^2 + 0.25s^2<BR>\n" );
document.write( "* L^2 = 10.5625s^2<BR>\n" );
document.write( "* L = s√10.5625<BR>\n" );
document.write( "* L = 2.84 * √10.5625 <BR>\n" );
document.write( "* L ≈ 9.31 inches\r<BR>\n" );
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document.write( "**7. Calculate the Lateral Surface Area**\r<BR>\n" );
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document.write( "* Lateral Surface Area = 4 * (1/2) * Base Edge * Slant Height<BR>\n" );
document.write( "* Lateral Surface Area = 4 * (1/2) * s * L<BR>\n" );
document.write( "* Lateral Surface Area = 4 * (1/2) * 2.84 * 9.31 <BR>\n" );
document.write( "* Lateral Surface Area ≈ 52.89 square inches\r<BR>\n" );
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document.write( "**Therefore, the lateral surface area of the square pyramid is approximately 53 square inches.**<BR>\n" );
document.write( " </SPAN>\n" );
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