I'm new to this site. I'm taking a differential equations course this summer and this question is from my first homework assignment. I was able to do all of my homework questions that just required me to solve the equations, but I have a hard time setting up equations from a word problem. I went to my TA's office hours this afternoon and he wasn't sure how to solve the problem, but recommended I try posting it on this site.
Here is the problem:
Consider a system of two tanks which hold salt-water. Fresh water flows into Tank 1. The well-mixed contents of Tank 1 flow into Tank 2. The well- mixed contents of Tank 2 flow out of the system. All three flow rates are 5 gal/min. Tank 1 initially holds 100gal of salt-water, Tank 2 initially holds 200gal, and each tank begins with 50lbs of salt.
(a) Find x(t), the amount of salt (in lbs) in Tank 1 at time t.
(b) Find the y(t), the amount of salt (in lbs) in Tank 2 at time t.
(c)Find the maximum amount of salt ever in Tank 2.
Thanks in advance! Also, I'm not necessarily looking for a solution; any tips you have towards an approach to the problem would be greatly appreciated!
Best Answer
Look at the first tank. How many salt flows out per minute? Since you have $x$ lbs of salt into $100$ gal and every minute 5 gal flow out, you should have: $$ x' = - \frac{5}{100}x. $$ Analogously: $$ y' = \frac{5}{100} x - \frac{5}{200} y. $$