00:01
Hello, the question deals with application of integrals.
00:04
We denote xt as amount of salt in the tank.
00:15
So the concentration of the solution in the tank is expressed as amount of salt that is 80kg divided by solution in the tank.
00:30
The tank which is thousand liters.
00:40
Now for the next value we formulate the required expression.
00:48
So we have d x by dt is equal to inflow rate that is 10 times the concentration inflow that is 0 .04 minus outflow rate that is 2 .04 minus outflow rate that is 10 times the concentration of outflow that is x t divided by 1000 plus 10 minus 10 t simplifying this we obtain d xy d t plus x t upon 100 is equal to 0 .4 now we solve this differential equation using integrating factor.
01:40
So i .f, that is the integrating factor is equals to e raised to the part 1 by 100 d t which is equals to e raised to the part t upon 100 and now the solution is expressed as x times the integrating factor e raised to the part t by 100 is equal to integration e raised to the power t upon 100 into 0 .4 dt plus c where c is integrating constant.
02:22
Simplifying this we obtain xt is equal to 40 plus c, e raised to the power minus t by 100.
02:36
Now the initial condition is x at 0 is equal to 80...