Algebra.Com's Answer #830090 by ikleyn(52800)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "A cone-shaped paper drinking cup is to hold 40cm^3 of water.
\n" );
document.write( "Find the height and radius of the cup that will require
\n" );
document.write( "the least amount of paper. NOTE: Enter the exact answers.
\n" );
document.write( "~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " The \"solution\" by @ewatrrr is INCORRECT.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " It is incorrect, since she uses the formula for the total surface area of the cone, \r
\n" );
document.write( "\n" );
document.write( " including its base, while in this problem the area of the base of the cone should NOT be included\r
\n" );
document.write( "\n" );
document.write( " and should not be considered, at all.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " So, I came to bring a correct solution.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "The volume of the cone cup is given 40 cm^3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The lateral area of a cone is\r\n" );
document.write( "\r\n" );
document.write( " S =  , (1)\r\n" );
document.write( "\r\n" );
document.write( "where r is the base radius and \"s\" is the slant height: s =  .\r\n" );
document.write( "\r\n" );
document.write( "So, we need minimize the lateral area\r\n" );
document.write( "\r\n" );
document.write( " S =  (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "at given restriction for the volume\r\n" );
document.write( "\r\n" );
document.write( "  = 40 cm^3. (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the restriction (3), \r\n" );
document.write( "\r\n" );
document.write( " h =  . \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We substitute it into expression (2), and we get S(r) as a function of the radius r, only\r\n" );
document.write( "\r\n" );
document.write( " S(r) =  =  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " +-------------------------------------------------------+\r\n" );
document.write( " | To find the minimum of S(r), we should calculate |\r\n" );
document.write( " | the derivative and equate it to zero. |\r\n" );
document.write( " +-------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "I will not calculate the derivative in full, which is a complicated fraction. \r\n" );
document.write( "It is enough to calculate its numerator and equate it to zero. It gives this equation\r\n" );
document.write( "\r\n" );
document.write( " - +  = 0,\r\n" );
document.write( "\r\n" );
document.write( "or, equivalently\r\n" );
document.write( "\r\n" );
document.write( " 7200 =  ,  =  , r =  = 3.0 cm. (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the radius is just found. The height should be \r\n" );
document.write( "\r\n" );
document.write( " h = =  = 4.24 cm. (5)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Expressions (4) and (5) give the required exact formulas and approximate numerical values.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "-------------------\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "For your safety, ignore the post by @ewatrrr, since her solution is incorrect.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "From day to day, this woman repeats the same error: she tries to solve problems that are out of her competency.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |