document.write( "Question 1012678: The volume,x litres of water present in a solution during a chemical process varies with time t seconds and satisfies the relation dx/dt=-3x/(1+t)^2. initially at t=0, x=1000. Show that at time t the volume is given by x=1000exp[-3t/(1+t)]. \n" ); document.write( "

Algebra.Com's Answer #628784 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"dx%2Fdt=-3x%2F%281%2Bt%29%5E2\"$
\n" ); document.write( " $\"dx%2Fx=-3%281%2Bt%29%5E%28-2%29%2Adt\"$
\n" ); document.write( "Use a substitution,
\n" ); document.write( " $\"u=1%2Bt\"$
\n" ); document.write( " $\"du=dt\"$
\n" ); document.write( "So,
\n" ); document.write( " $\"dx%2Fx=-3u%5E%28-2%29du\"$
\n" ); document.write( "Integrating both sides,
\n" ); document.write( " $\"ln%28x%29=3u%5E%28-1%29%2BC\"$
\n" ); document.write( " $\"ln%28x%29=3%2F%281%2Bt%29%2BC\"$
\n" ); document.write( " $\"x=Ce%5E%283%2F%281%2Bt%29%29\"$
\n" ); document.write( "When $\"t=0\"$ , $\"x=1000\"$
\n" ); document.write( " $\"1000=Ce%5E%283%2F%281%2B0%29%29\"$
\n" ); document.write( " $\"C=1000%2Fe%5E3\"$
\n" ); document.write( " $\"x=%281000%2Fe%5E3%29%2Ae%5E%283%2F%281%2Bt%29%29\"$
\n" ); document.write( "Check your problem setup because our functions don't match.\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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