A tank contains 70 kg of salt and 2000 L of water. Water containing 0.4kg/L of...

Pg 1 solution Initially tank contains 70 kyy Salt and 2000 L Alt is the amount salt in the tank at time t measured in erams.

Of water. kilograms. at (initially) , donk contains 70 kg ho tas. 918) 40 kg COI b Let's the tank librated evention de quation. contains (70 kg of salt hence the initial condition is A (0) = 70 kg The di differential equation in A'tho da = reate in rate out The rate will be * rate i.e, Concentration of the form amount of salt (kg ofy water (L) rate (l/min) amount GE on N B612

Rake lg 2 Notulo Liquid coming in, water containing 0.4 kg/L of salt is coming in the tank at the 16 L/min.

0.4kg /L* 16 L/ min rate of Rate in = * 6.4 kg /min Rate the tank going out: The solution drains from the toplo, ale tohle to get The amount of solution is also ahalying with respect to time. solution is coming in at 164/min and a at 4 L/min hence after every min we Liquid solution is leaving have tank. extra 1210 of The amount cat time t i then of liquid 2000 tlat Thus the rate out is Akg (20007 lat) L 1x (4 L/min) GA 2000 tlat kg/min B612

193 Ikus our initial value problem is A' = rate in -rate out det so imin 4A 2000 + lat, A 6.4 OA 1000 t 6t Goth salt where A is rate in the satt tank of change of one consumers kilograma A is the amount salt at time to А + QA losotot 6.4=0 Answer A + A 2 1000+6t comparing this equation with standard equation of Caterintegrating factor + f(x) y = Q(x) method equation. da where P(x) f Q(x) be a & may are functions constants. o 3612

194 Integrating factor (IF) at 1000 tot If = е.

1000+ 6t If e 2. ! ln 1000 + 6t IA 6 e 7 ln 1000 +67 1 If = e Answer If= (1000 +67) 73 by log properties (2) The colution of equation i) is If A- If. Q at (1000 + 6t) A= | (100+ 6.4 1.6.4 dt us (1000+ 6+) A 6.4 (1000+ 6+) 13 413 (loopt 6+) A= 1000+ 6t) tc at t=0 A (0) = 70 kg B612

to. A 13 (1000 to yo 70 = 0:8 (1000+ 0 tc 10x7o = 0.8 x 10000 tc C= 7300 A = (1000+ 6+ / A = 0.8/1080 t6 ) 3 - 4300 0.8 (1000 + 6+) - 7300(1070 +60's 730(100+ 607's Dble the amount of salt in the kilograme. A = 4.8t tsoo tank at time t measured in ki le the concentration salt in the tank as time approaches.

Importy into The put to . Alt) = 4.8t + Box --309 (1ooot67) kg concentration of salt in the solukon. in the tank approacher as time approaches infinity . B612 infinite