document.write( "Question 1110914: An open box, without a top is to be made by cutting a congruent squares from each corner of a rectangular sheet metal, which measures 5 inches by 8 inches, and folding up the sides. Find the volume of the box, which has the greatest volume. \n" ); document.write( "

Algebra.Com's Answer #725926 by ikleyn(52787) $\"\"$ $\"About$

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

\r\n" );
document.write( "Cutting the (h x h)-squares from the 5X8 inches metal sheet and folding, you get the rectangular prism (open box) of dimensions\r\n" );
document.write( "\r\n" );
document.write( "    (5-2h) x (8-2h) x h\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "with the volume of   V(h)= h*(5-2h)*(8-2h) = 4h^3 -26h^2 + 40h cubic inches.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To find the maximal volume, take the derivative\r\n" );
document.write( "\r\n" );
document.write( " = 12h^2 - 52h + 40\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and equate it to zero:\r\n" );
document.write( "\r\n" );
document.write( "12h^2 - 52h + 40 = 0,\r\n" );
document.write( "\r\n" );
document.write( "3h^2 - 13h + 10 = 0,\r\n" );
document.write( "\r\n" );
document.write( " =  = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "There are two roots:   =  =  =   and   =  = 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Compare the values\r\n" );
document.write( "\r\n" );
document.write( "    V((10/3) =  = -7.4\r\n" );
document.write( "\r\n" );
document.write( "and\r\n" );
document.write( "\r\n" );
document.write( "    V(1)     =  = 18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The answer is  h = 1.\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" ); document.write( "

\n" );