document.write( "Question 1119872: A rectangular box with a square base and open top is to be made. Find the volume of the largest box that can be made from 432 sq. m. of material. \n" ); document.write( "

Algebra.Com's Answer #735534 by greenestamps(13209) $\"\"$ $\"About$

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\n" ); document.write( "Let the side of the square base be x and the height be h. Then

\n" ); document.write( "(1) the total surface area (base and 4 sides) in square meters is 432:
\n" ); document.write( " $\"x%5E2%2B4xh+=+432\"$

\n" ); document.write( "(2) the volume (to be maximized) is the area of the base times the height:
\n" ); document.write( " $\"V+=+x%5E2h\"$

\n" ); document.write( "Solve equation (1) for h and substitute in equation (2) to get an expression for the volume in terms of the single variable x:

\n" ); document.write( " $\"4xh+=+432-x%5E2\"$
\n" ); document.write( " $\"h+=+%28432-x%5E2%29%2F4x\"$
\n" ); document.write( " $\"V+=+x%5E2%28%28432-x%5E2%29%2F4x%29+=+%28432x%5E2-x%5E4%29%2F4x+=+108x-x%5E3%2F4\"$

\n" ); document.write( "Take the derivative and set it equal to 0 to find the value of x that maximizes the volume; and calculate the volume for that value of x:

\n" ); document.write( " $\"108+-+%283%2F4%29x%5E2+=+0\"$
\n" ); document.write( " $\"%283%2F4%29x%5E2+=+108\"$
\n" ); document.write( " $\"x%5E2+=+144\"$
\n" ); document.write( " $\"x+=+12\"$
\n" ); document.write( " $\"V+=+108%2812%29-%2812%5E3%29%2F4+=+1296-432+=+864\"$

\n" ); document.write( "The maximum volume is 864 cubic meters, when the square base is 12m on a side and the height is 6m. \n" ); document.write( "

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