lake with volume V=100 km^3 is fed by a river at a rate of r km^3 /yr. In addition...

[mfm12a]

1. A lake with volume V = 100 km3, is fed by a river at a rate of r km3/yr. In addition, there is afactory on the lake that introduces a pollutant into the lake at the rate of p km3/yr. There is anotherriver that is fed by the lake at a rate that keeps the volume of the lake constant. This means thatthe rate of flow from the lake into the outlet river is (p + r) km3/yr. Let x(t) denote the volume ofthe pollutant in the lake at time t. Then c(t) = x(t)/V is the concentration of the pollutant.
   

1. 
   a) Under the assumption of immediate and perfect mixing of the pollutant into the lake water,show that the concentration satisfies the differential equation
   
   c' + ((p+r)/V)= (p/V).
   
   b) It has been determined that a concentration of over 2% is hazardous for fish in the lake. Supposethat r = 50 km3/yr, p = 2 km3/yr, and the initial concentration of pollutant in the lake is zero.How long will it take the lake to become hazardous to the health of the fish? You may use Maplefor your analysis on this problem, but provide your answers below.
   

 

[Deleted member 4993]

2. A lake with volume V = 100 km3, is fed by a river at a rate of r km3/yr. In addition, there is afactory on the lake that introduces a pollutant into the lake at the rate of p km3/yr. There is anotherriver that is fed by the lake at a rate that keeps the volume of the lake constant. This means thatthe rate of flow from the lake into the outlet river is (p + r) km3/yr. Let x(t) denote the volume ofthe pollutant in the lake at time t. Then c(t) = x(t)/V is the concentration of the pollutant.

2. 
   a) Under the assumption of immediate and perfect mixing of the pollutant into the lake water,show that the concentration satisfies the differential equation
   
   c' + ((p+r)/V)= (p/V).
   
   b) It has been determined that a concentration of over 2% is hazardous for fish in the lake. Supposethat r = 50 km3/yr, p = 2 km3/yr, and the initial concentration of pollutant in the lake is zero.How long will it take the lake to become hazardous to the health of the fish? You may use Maplefor your analysis on this problem, but provide your answers below.

You have posted ~9 problems without showing one line of original work!!

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33