document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198563>Question 1198563</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The bases of a frustum of a right circular cone are circles whose diameters are respectively 18 and 14 inches. With a slant height of 25 inches, the volume may be expressed as V = Χπ √γ in^3 where Χ and γ are integers. Find the smallest sum of Χ and γ. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848324 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=848324\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Certainly, let's find the volume of the frustum and determine the smallest sum of Χ and γ.\r<BR>\n" );
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document.write( "**1. Find the radii of the bases:**\r<BR>\n" );
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document.write( "* Radius of the larger base (R): 18 inches / 2 = 9 inches<BR>\n" );
document.write( "* Radius of the smaller base (r): 14 inches / 2 = 7 inches\r<BR>\n" );
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document.write( "**2. Find the height (h) of the frustum:**\r<BR>\n" );
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document.write( "* We can use the Pythagorean theorem with the slant height (l = 25 inches), the radius difference (R - r = 2 inches), and the height (h):\r<BR>\n" );
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document.write( "   l² = (R - r)² + h²<BR>\n" );
document.write( "   25² = 2² + h²<BR>\n" );
document.write( "   h² = 625 - 4<BR>\n" );
document.write( "   h² = 621<BR>\n" );
document.write( "   h = √621 inches\r<BR>\n" );
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document.write( "**3. Calculate the volume (V) of the frustum:**\r<BR>\n" );
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document.write( "* The formula for the volume of a frustum of a right circular cone is:\r<BR>\n" );
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document.write( "   V = (1/3) * π * h * (R² + Rr + r²) \r<BR>\n" );
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document.write( "* Substitute the values:\r<BR>\n" );
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document.write( "   V = (1/3) * π * √621 * (9² + 9*7 + 7²)<BR>\n" );
document.write( "   V = (1/3) * π * √621 * (81 + 63 + 49)<BR>\n" );
document.write( "   V = (1/3) * π * √621 * 193\r<BR>\n" );
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document.write( "**4. Express the volume in the form V = Χπ√γ**\r<BR>\n" );
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document.write( "* V = (1/3) * π * √621 * 193 <BR>\n" );
document.write( "* V = (193/3) * π * √621\r<BR>\n" );
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document.write( "* Therefore: <BR>\n" );
document.write( "    * Χ = 193<BR>\n" );
document.write( "    * γ = 621\r<BR>\n" );
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document.write( "**5. Find the smallest sum of Χ and γ**\r<BR>\n" );
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document.write( "* Χ + γ = 193 + 621 = 814\r<BR>\n" );
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document.write( "**Therefore, the smallest sum of Χ and γ is 814.**<BR>\n" );
document.write( "**1. Find the Radii**\r<BR>\n" );
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document.write( "* **Radius of the larger base:** 18 inches / 2 = 9 inches<BR>\n" );
document.write( "* **Radius of the smaller base:** 14 inches / 2 = 7 inches\r<BR>\n" );
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document.write( "**2. Find the Height of the Frustum**\r<BR>\n" );
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document.write( "* Let 'h' be the height of the frustum.<BR>\n" );
document.write( "* We can use the Pythagorean theorem with the slant height and the difference in radii to find the height.\r<BR>\n" );
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document.write( "* Consider a right triangle formed by:<BR>\n" );
document.write( "    * The base: Difference in radii = 9 inches - 7 inches = 2 inches<BR>\n" );
document.write( "    * The height: 'h'<BR>\n" );
document.write( "    * The hypotenuse: Slant height = 25 inches\r<BR>\n" );
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document.write( "* Using Pythagorean Theorem:<BR>\n" );
document.write( "    * h² = 25² - 2²<BR>\n" );
document.write( "    * h² = 625 - 4<BR>\n" );
document.write( "    * h² = 621<BR>\n" );
document.write( "    * h = √621 inches\r<BR>\n" );
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document.write( "**3. Find the Volume of the Frustum**\r<BR>\n" );
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document.write( "* The formula for the volume (V) of a frustum of a right circular cone is:\r<BR>\n" );
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document.write( "   V = (1/3) * π * h * (R² + Rr + r²) \r<BR>\n" );
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document.write( "   where:<BR>\n" );
document.write( "      * h is the height of the frustum <BR>\n" );
document.write( "      * R is the radius of the larger base (9 inches)<BR>\n" );
document.write( "      * r is the radius of the smaller base (7 inches)\r<BR>\n" );
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document.write( "* V = (1/3) * π * √621 * (9² + 9*7 + 7²) <BR>\n" );
document.write( "* V = (1/3) * π * √621 * (81 + 63 + 49) <BR>\n" );
document.write( "* V = (1/3) * π * √621 * 193\r<BR>\n" );
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document.write( "**4. Express the Volume in the Form V = Χπ√γ**\r<BR>\n" );
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document.write( "* V = (193/3) * π * √621 \r<BR>\n" );
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document.write( "* **Therefore:**<BR>\n" );
document.write( "    * Χ = 193/3 <BR>\n" );
document.write( "    * γ = 621\r<BR>\n" );
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document.write( "**5. Find the Smallest Sum of Χ and γ**\r<BR>\n" );
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document.write( "* Sum = Χ + γ = (193/3) + 621 <BR>\n" );
document.write( "* Sum = (193 + 1863)/3 <BR>\n" );
document.write( "* Sum = 2056/3 \r<BR>\n" );
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document.write( "**Therefore, the smallest sum of Χ and γ is 2056/3.**<BR>\n" );
document.write( " </SPAN>\n" );
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