document.write( "Question 30833: There is a box with a square base, and a volume of 32000cm^3, with an open top, what is the least material used, and what are the dimensions? \n" ); document.write( "

Algebra.Com's Answer #17583 by josmiceli(19441) $\"\"$ $\"About$

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Let a side of the square base = x
\n" ); document.write( "Let the height of the box = y
\n" ); document.write( "The volume, V =
\n" ); document.write( " $\"V+=+32%2A10%5E3\"$
\n" ); document.write( " $\"32%2A10%5E3+=+x%5E2+%2A+y\"$
\n" ); document.write( "The dimensions of the materials = 4*x*y for the sides
\n" ); document.write( "and x^2 for the base
\n" ); document.write( "M = materials
\n" ); document.write( " $\"M+=+4%2Ax%2Ay+%2B+x%5E2\"$
\n" ); document.write( "from above
\n" ); document.write( " $\"y+=+32%2A10%5E3+%2F+x%5E2\"$
\n" ); document.write( " $\"M+=+4+%2A+x+%2A+%2832%2A10%5E3%2F+x%5E2%29+%2B+x%5E2\"$
\n" ); document.write( " $\"M+=+4+%2A+%2832%2A10%5E3%2F+x%29+%2B+x%5E2\"$
\n" ); document.write( "find M'= MP the dirivative of M
\n" ); document.write( " $\"MP+=+32%2A10%5E3%2A+%28-4%2Ax%5E%28-2%29%29+%2B+2+%2A+x\"$
\n" ); document.write( "set MP = 0 to find a minimum
\n" ); document.write( " $\"0+=+32%2A10%5E3%2A+%28-4%2Ax%5E%28-2%29%29+%2B+2+%2A+x\"$
\n" ); document.write( " $\"%2832+%2A+10%5E3+%2A+4%29+%2F+x%5E2+=+2%2A+x\"$
\n" ); document.write( "multiply both sides by x^2
\n" ); document.write( " $\"x%5E3+=+4+%2A+32+%2A+10%5E3\"$
\n" ); document.write( " $\"x%5E3+=+64+%2A+10%5E3\"$
\n" ); document.write( " $\"x+=+40\"$
\n" ); document.write( "The base is 40X40 cm
\n" ); document.write( " $\"y+=+32%2A10%5E3+%2F+x%5E2\"$
\n" ); document.write( " $\"y+=+32%2A10%5E3+%2F+40%5E2\"$
\n" ); document.write( " $\"y+=+32+%2A+10%5E3+%2F+16+%2A+10%5E2\"$
\n" ); document.write( " $\"y+=+20\"$
\n" ); document.write( "The height is 20 cm\r
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