document.write( "Question 1195557: A candy company wants to reduce the amount of packaging they use, by minimizing the
\n" ); document.write( "surface area. If they still want the candy package to hold a volume of 250 cm3, what is the
\n" ); document.write( "minimum surface area? Assume the candy comes packaged in rectangular prisms. Round to
\n" ); document.write( "one decimal place. \n" ); document.write( "

Algebra.Com's Answer #828084 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A candy company wants to reduce the amount of packaging they use, by minimizing the
\n" ); document.write( "surface area. If they still want the candy package to hold a volume of 250 cm3, what is the
\n" ); document.write( "minimum surface area? Assume the candy comes packaged in rectangular prisms. Round to
\n" ); document.write( "one decimal place.
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document.write( "Let x, y and z be the dimensions of a rectangulas prism.\r\n" );
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document.write( "Then its volume is  \r\n" );
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document.write( "    xyz = 250 cm^3,     (1)\r\n" );
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document.write( "a fixed value, and the problem wants we minimize the surface area  2xy + 2yz + 2xz  under this restriction (1).\r\n" );
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document.write( "It is the same as to minimize the function of 3 independent variables\r\n" );
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document.write( "    F(x,y,z) = xy + yz + xz\r\n" );
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document.write( "under restriction (1).\r\n" );
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document.write( "Using the restriction, we can reduce function F(x,y,z) to function of two independent variables\r\n" );
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document.write( "    f(x,y) = xy +  + \r\n" );
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document.write( "and look for the minimum of this function.\r\n" );
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document.write( "To find its minimum, we take partial derivatives of  f(x,y)  over x and y  and equate them to zero.\r\n" );
document.write( "It gives us this system of equations\r\n" );
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document.write( "    y -  = 0,    (2)     (x-derivative)\r\n" );
document.write( "  \r\n" );
document.write( "    x -  = 0.    (3)     (y-derivative)\r\n" );
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document.write( "From (2) and (3)\r\n" );
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document.write( "    x^2*y = 250    (4)\r\n" );
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document.write( "    x*y^2 = 250.   (5)\r\n" );
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document.write( "Dividing (4) by (5), we get\r\n" );
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document.write( "     = 1,  or  x = y.\r\n" );
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document.write( "Working similarly with the other pair of independent variables, we can get similarly  x = z,\r\n" );
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document.write( "which tells us that the minimum surface value is achieved for the cube x = y = z.\r\n" );
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document.write( "Its dimension is   = 250,  or  x =  = 6.3  (rounded to one decimal place).\r\n" );
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document.write( "ANSWER.  The minimum surface area rectangular prism is a cube with the edge size of   = 6.30 cm.\r\n" );
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document.write( "         Its surface area is  = 238.1 cm^2  (rounded to one decimal place).\r\n" );
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document.write( "CHECK.   = 250.047 cm^3.\r\n" );
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