document.write( "Question 1164246: The numbers 1,2,3,4,5,6,7,8,9,10 are to be entered into the 10 boxes shown below, so that each number is used exactly once:
\n" ); document.write( "P = (blank + blank + blank + blank+ blank)(blank + blank + blank + blank + blank)
\n" ); document.write( "What is the maximum value of P? What is the minimum value of P?\r
\n" ); document.write( "\n" ); document.write( "Blank stands for the empty boxes that were in the original problem. \n" ); document.write( "
Algebra.Com's Answer #788544 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Think of \r\n" );
document.write( "\r\n" );
document.write( "(blank + blank + blank + blank+ blank)(blank + blank + blank + blank + blank)\r\n" );
document.write( "\r\n" );
document.write( "as finding the area of a rectangle A = LW.\r\n" );
document.write( "\r\n" );
document.write( "Think of the set of rectangles with integer sides such that each rectangle\r\n" );
document.write( "has its length and width together made up of 10 segments of sizes\r\n" );
document.write( "1,2,3,...,10. So its perimeter = 2L+2W 2(L+W)= 2(1+...+10) = 110.\r\n" );
document.write( "So our problem is a finite subset of the set of areas of the infinite set of\r\n" );
document.write( "rectangles with perimeter 110.  We know that the area of the largest\r\n" );
document.write( "possible rectangle is a square with sides 1/4 of 110 or 27.5.  So to find\r\n" );
document.write( "the maximum area in our subset of areas we need to find a rectangle which is\r\n" );
document.write( "as close as possible to a square with sides 27.5. \r\n" );
document.write( "\r\n" );
document.write( "So I'll see if it's possible to get one dimension to be 27 and the other to\r\n" );
document.write( "be 28, then they'll be closest together.  I'll start with the largest\r\n" );
document.write( "possible two (10+9) and the smallest possible two (1+2) and get\r\n" );
document.write( "(10+9+1+2)=22, so if I add 5 to that I'll have 10+9+5+1+2=27.  Then the\r\n" );
document.write( "other five will be 8+7+6+4+3=28.  Hurray! So I claim that the largest\r\n" );
document.write( "product is 27*28 = 756.  For the two can be no closer in area to the area of\r\n" );
document.write( "the square with maximum area, which is 27.5 x 27.5.\r\n" );
document.write( "  \r\n" );
document.write( "So the maximum product possible is (10+9+5+1+2)(8+7+6+4+3) = (27)(28) = 756.\r\n" );
document.write( "\r\n" );
document.write( "The minimum area of a rectangle with circumference 110 when one\r\n" );
document.write( "of the two dimensions is 0 and the other is 55, which is a rectangle\r\n" );
document.write( "degenerated into a line segment.  That is when the two dimensions are as far\r\n" );
document.write( "apart as possible.\r\n" );
document.write( "\r\n" );
document.write( "So the minimum product is when one factor is as small as possible and the\r\n" );
document.write( "other is as large as possible.  That's this case:\r\n" );
document.write( "\r\n" );
document.write( "(1+2+3+4+5)(6+7+8+9+10) = (15)(40) = 600\r\n" );
document.write( "\r\n" );
document.write( "So 600 is the minimum product\r\n" );
document.write( "\r\n" );
document.write( "Answers: the maximum value of P = 756 (when the two numbers are closest\r\n" );
document.write( "together, (27)(28) = 756\r\n" );
document.write( "         the minimum value of P = 600 (when the two numbers are farthest\r\n" );
document.write( "apart, (15)(40) = 600 \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );