# Math Help - modeling first order equation..(Long question, fairly easy tho)

1. ## modeling first order equation..(Long question, fairly easy tho)

A tank originally contains 100 gal of fresh water. Then water containing 0.5 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After ten minutes the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min.

im getting the part up to the first 10 minutes, but im getting stuck on the second part which starts up again without salt being poured in.

the first DE I got for the rate with salt poured in was $50(1-e^{-.2})$ so that equation satisfies how much salt is left after the first 10 min exchange is made. Which seems about right and gives me 9.066 lbs of salt left. Im just not sure how to continue passed that.

more generally my equation was $50 + (Qo - 50)e^{-rt/100}$

2. Originally Posted by p00ndawg
A tank originally contains 100 gal of fresh water. Then water containing 0.5 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After ten minutes the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min.

im getting the part up to the first 10 minutes, but im getting stuck on the second part which starts up again without salt being poured in.

the first DE I got for the rate with salt poured in was $50(1-e^{-.2})$ so that equation satisfies how much salt is left after the first 10 min exchange is made. Which seems about right and gives me 9.066 lbs of salt left. Im just not sure how to continue passed that.

more generally my equation was $50 + (Qo - 50)e^{-rt/100}$
Set up a new ODE

First problem

$\frac{dA}{dt} = r_i c_i - r_i c_o = 2 \cdot \frac{1}{2} - 2 \cdot \frac{A}{100},\;\;\;A(0) = 0
$

Second problem

$\frac{dA}{dt} = r_i c_i - r_i c_o = 2 \cdot 0 - 2 \cdot \frac{A}{100},\;\;\;A(0) = 9.066$

3. Originally Posted by danny arrigo
Set up a new ODE

First problem

$\frac{dA}{dt} = r_i c_i - r_i c_o = 2 \cdot \frac{1}{2} - 2 \cdot \frac{A}{100},\;\;\;A(0) = 0
$

Second problem

$\frac{dA}{dt} = r_i c_i - r_i c_o = 2 \cdot 0 - 2 \cdot \frac{A}{100},\;\;\;A(0) = 9.066$