document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178404>Question 1178404</A>: <I><SPAN STYLE=\"word-break: break-all;\"> in a laboratory, a chemist's measuring glass is conical in shape. if it is 8cm deep and 3 cm across the mouth, find the distance on the slant edge between the markings for 1 cc and 2 cc. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850349 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850349\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Visualize the Cone**\r<BR>\n" );
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document.write( "Imagine a cone representing the measuring glass.\r<BR>\n" );
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document.write( "* **Height (h):** 8 cm<BR>\n" );
document.write( "* **Diameter (d):** 3 cm<BR>\n" );
document.write( "* **Radius (r):** d/2 = 3/2 = 1.5 cm\r<BR>\n" );
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document.write( "**2. Volume of a Cone**\r<BR>\n" );
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document.write( "The volume (V) of a cone is given by:\r<BR>\n" );
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document.write( "* V = (1/3)πr²h\r<BR>\n" );
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document.write( "**3. Similar Cones**\r<BR>\n" );
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document.write( "When we fill the cone to different levels, we create smaller cones that are similar to the larger cone. This means their dimensions are proportional.\r<BR>\n" );
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document.write( "**4. Finding the Radii and Heights for 1 cc and 2 cc**\r<BR>\n" );
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document.write( "Let's denote:\r<BR>\n" );
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document.write( "* r1, h1: radius and height for 1 cc<BR>\n" );
document.write( "* r2, h2: radius and height for 2 cc\r<BR>\n" );
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document.write( "We know:\r<BR>\n" );
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document.write( "* 1 = (1/3)πr1²h1<BR>\n" );
document.write( "* 2 = (1/3)πr2²h2\r<BR>\n" );
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document.write( "Also, by similarity:\r<BR>\n" );
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document.write( "* r1/h1 = r/h = 1.5/8<BR>\n" );
document.write( "* r2/h2 = r/h = 1.5/8\r<BR>\n" );
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document.write( "From these proportions, we have:\r<BR>\n" );
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document.write( "* r1 = (1.5/8)h1<BR>\n" );
document.write( "* r2 = (1.5/8)h2\r<BR>\n" );
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document.write( "Substitute these into the volume equations:\r<BR>\n" );
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document.write( "* 1 = (1/3)π((1.5/8)h1)²h1 = (1/3)π(2.25/64)h1³<BR>\n" );
document.write( "* 2 = (1/3)π((1.5/8)h2)²h2 = (1/3)π(2.25/64)h2³\r<BR>\n" );
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document.write( "Solve for h1 and h2:\r<BR>\n" );
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document.write( "* h1³ = (64 * 3) / (2.25π) ≈ 27.147<BR>\n" );
document.write( "* h1 ≈ ∛27.147 ≈ 3.003 cm<BR>\n" );
document.write( "* h2³ = (64 * 6) / (2.25π) ≈ 54.294<BR>\n" );
document.write( "* h2 ≈ ∛54.294 ≈ 3.784 cm\r<BR>\n" );
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document.write( "Now find r1 and r2:\r<BR>\n" );
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document.write( "* r1 = (1.5/8) * 3.003 ≈ 0.563 cm<BR>\n" );
document.write( "* r2 = (1.5/8) * 3.784 ≈ 0.710 cm\r<BR>\n" );
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document.write( "**5. Finding the Slant Edge Distances**\r<BR>\n" );
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document.write( "Let:\r<BR>\n" );
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document.write( "* s1: slant edge distance for 1 cc<BR>\n" );
document.write( "* s2: slant edge distance for 2 cc\r<BR>\n" );
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document.write( "Using the Pythagorean theorem (in 3D):\r<BR>\n" );
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document.write( "* s1 = √(r1² + h1²) = √(0.563² + 3.003²) ≈ √(0.317 + 9.018) ≈ √9.335 ≈ 3.055 cm<BR>\n" );
document.write( "* s2 = √(r2² + h2²) = √(0.710² + 3.784²) ≈ √(0.504 + 14.319) ≈ √14.823 ≈ 3.850 cm\r<BR>\n" );
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document.write( "**6. Finding the Distance Between Markings**\r<BR>\n" );
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document.write( "The distance on the slant edge between the 1 cc and 2 cc markings is:\r<BR>\n" );
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document.write( "* s2 - s1 ≈ 3.850 - 3.055 ≈ 0.795 cm\r<BR>\n" );
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document.write( "**Therefore, the distance on the slant edge between the markings for 1 cc and 2 cc is approximately 0.795 cm.**<BR>\n" );
document.write( " </SPAN>\n" );
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