document.write( "Question 1190223: a besieged fortress is held by 5700 men who have provisions for 66 days. of the garrison loses 20 men each day, for how many days will be provisions last? \n" ); document.write( "

Algebra.Com's Answer #821836 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "a besieged fortress is held by 5700 men who have provisions for 66 days.
\n" ); document.write( "of the garrison loses 20 men each day, for how many days will be provisions last?
\n" ); document.write( "~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "There are 5700*66 portions of provision in the storage.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In the 1st day, 5700 portions are consumed.\r\n" );
document.write( "\r\n" );
document.write( "In the 2nd day, 5700-20 = 5680 portions are consumed.\r\n" );
document.write( "\r\n" );
document.write( "In the 3nd day, 5680-20 = 5660 portions are consumed.\r\n" );
document.write( "\r\n" );
document.write( ". . . and so on . . . \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The problem wants you find the sum of the first n terms of this arithmetic progression\r\n" );
document.write( "\r\n" );
document.write( "     = \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and the number n in such a way that  = 5700*66, under an additional condition  >= 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The first term of this AP is 5700; the common difference is -20,  so the sum of the first n terms is\r\n" );
document.write( "\r\n" );
document.write( "     =  =  = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So we write this equation\r\n" );
document.write( "\r\n" );
document.write( "     = 5700*66\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Simplify \r\n" );
document.write( "\r\n" );
document.write( "    20n^2 - 11420n + 11400*66 = 0\r\n" );
document.write( "\r\n" );
document.write( "      n^2 - 571 + 5700*66 = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, apply the quadratic formula.  You will get\r\n" );
document.write( "\r\n" );
document.write( "       = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The value  =  = 76 works: it satisfies  =  = 5700*66 and  >= 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The other value  =  = 495 does not work:  = 5700-20*495 = -4200 is negative.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the problem is just solved, and the  ANSWER  is 76 days.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To check, I created an MS Excel spreadsheet, generated this AP there and calculated its sum.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The check confirmed that the answer is correct.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );