document.write( "Question 1155393: A box with a square bottom and no top is to be made to contain 100 cubic inches. Bottom material costs five cents per square and side material costs two cents per square inch. Find the cost of least expensive box that can be made.\r
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Algebra.Com's Answer #778030 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "Let x be the size of the squared base edge and h be the height.\r\n" );
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document.write( "Then the volume is  x^2*h = 100  cubic inches \r\n" );
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document.write( "and the lateral surface area is 4*xh square inches.\r\n" );
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document.write( "The total cost is  C(x,y) = 5x^2 + 2*4*xh = 5x^2 + 8xh.\r\n" );
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document.write( "So we need to minimize  C(x,y) = 5x^2 + 8xh  under restriction x^2*h = 100.\r\n" );
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document.write( "We then express h =  and substitute it into the expression for C(x,y).\r\n" );
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document.write( "We then get\r\n" );
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document.write( "    C(x,y) =  +  =  + ,\r\n" );
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document.write( "and we need to minimize this function.\r\n" );
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document.write( "Find the derivative and equate it to zero\r\n" );
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document.write( "    10x -  = 0.\r\n" );
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document.write( "From this equation, find x\r\n" );
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document.write( "    10*x^3 = 800\r\n" );
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document.write( "    x^3 =  = 80.\r\n" );
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document.write( "    x =  = 4.309  inches.\r\n" );
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document.write( "Then h =  =  = 5.386 inches.\r\n" );
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document.write( "ANSWER.  The base size is 4.309 inches;  the height is 5.386 inches.\r\n" );
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\n" ); document.write( "\n" ); document.write( "If you want to see many other similar solved problems, look into the lesson\r
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