document.write( "Question 1116208: A cereal box is designed to hold 3375 cubic cm of cereal. What dimensions for the box will minimize the cost of producing the box? \n" ); document.write( "

Algebra.Com's Answer #731051 by ikleyn(52794) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "I don't know about the cost of producing the box, and I think that this passage is not relevant to the rest of the condition.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "But I know which dimensions will minimize the surface area of the box (which directly relate to the cost of the material).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The dimensions what minimize the surface area are 15 x 15 x 15 centimeters:  the box must be a cube.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is easy to get this result analytically, using Calculus.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The surface area of the (x,y,z)-box is  A(x,y,z) = 2*(xy + xz + yz).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The volume = xyz = 3375,  so  z = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, A(x,y,z) =   at the given volume.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The conditions   =  = 0 give\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     -  = 0  ====>  x^3 = 3375  ====>  x =  = 15,   and\r\n" );
document.write( "\r\n" );
document.write( "     -  = 0  ====>  y^3 = 3375  ====>  y =  = 15.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "And then  z =  =  = 15.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );