Question

g V. (10 points) Suppose that a large tank initially holds 400 gallons of brine, which contains 20 pounds of salt. Pure water is pumped into the tank at a rate of 5 gallons per minute. When the solution in the tank is well stirred, it is pumped out at a rate of 4 gallons per minute. a. Write the initial value problem that represents the change in the amount of salt at time t. b. Solve the initial value problem.

Answer 1

Answer:

Step-by-step explanation:

If you assume concentration of salt per unit water remains constnat, find the constant.

Initially, for 400 gal of water, you have 20 pounds of salt

20lb/400gal=0.05lb salt per gal of water.

Now do part a)

I will use the following variables:

$y(t)$: lb of salt in tank at time t

$o(t)$: lb of salt pumped out at time t

$i(t)$: lb of salt pumped in at time t

$a$: lb of salt initially in tank

the equation is

total salt=initial salt+salt in-salt out

$y(t)=a+i(t)-o(t)$

we know that $a$ is 20lb

$i(t)$ is 5 gal per minute, but we want lbs salt so 5gal/min*0.05lb/gal=0.25lb/min. but this is per time, so multiply by t to get 0.25t

$o(t)$ is 4gal per minute, but we want lbs salt so 3gal/min*0.05lb/gal=0.20lb/min. but this is per time, so multiply to get 0.20t

subsitute all in

$y(t)=a+i(t)-o(t)$

$y(t)=20+0.25t-0.20t$

$y(t)=20+0.05t$

part b)

to solve this initial value problem, I would need to be given either a goal amount of salt ($y(t)$) or a time elapsed $t$. Since none are given, the IVP cannot be solved.