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 #628801 by fractalier(6550) $\"\"$ $\"About$

You can put this solution on YOUR website!
From dx/dt=-3x/(1+t)^2, I'm thinking to solve this by separation of variables...thus, we can rearrange and get
\n" ); document.write( "(1/x) dx = -3 dt / (t+1)^2
\n" ); document.write( "Now integrate
\n" ); document.write( "ln x = 3/(t+1) + C
\n" ); document.write( "Now exponentiate
\n" ); document.write( "x(t) = Ce^(3/(t+1))
\n" ); document.write( "Now apply initial conditions...
\n" ); document.write( "x(0) = 1000 = Ce^3 so that
\n" ); document.write( "C = 1000/e^3 = 1000e^(-3)
\n" ); document.write( "and then
\n" ); document.write( "x(t) = 1000e^(-3)*e^(3/(t+1))
\n" ); document.write( "Now combine exponents and you get
\n" ); document.write( "x(t) = 1000e^(-3t/(1+t)) \n" ); document.write( "

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