Question

Consider a pond that initially contains 10 million gallons of fresh water. Water containing a chemical pollutant flows into the pond at the rate of 5 million gallons per year (gal/yr), and the mixture in the pond flows out at the same rate. The concentration c=c(t) of the chemical in the incoming water varies periodically with time to the expression c(t) = 2 + sin(2t) grams per gallon (g/gal).
Construct a mathematical model of this flow process and determine the amount of chemical in the pond at any time t. Then, plot the solution using Maple and describe in words the effect of the variation in the incoming chemical.

Answer 1

Answer / Explanation:

To solve this , we start by recalling the conversion principle:

Thus,  dc/dt = Rate of C in - Rate of C out,

Therefore, for the number of chemical C(t) , i gram, in the pond at time (t), the rate of flow of C is modeled as:

Rate of Cin = Cin F,

Where Cin is the concentration of the chemical going in ( in this  case, Cin = 2 grams per gallon), and F is the rate of flow of water  into the pond (in this case, F = 5 million gallons per year). We  then have that

          Rate of C in = 10 million grams per year

Therefore the rate of Cout = c(t)F, where  c(t) = C(t) / V is the concentration of the chemical in the pond ( here, we are  assuming instant mixing). The volume, (V)  of the water in the  pond is 10 million gallons (we are assuming that the rate of flow  of water into the pond is the same as the rate of flow out, so that  the volume of water in the pond remains constant). Thus,

           Rate of C  out = 1/2 C(t) million grams per year.

Thus, the differential equation describing the evolution of C =

C(t) in time is: dc/dt = 10 - C/2 in millions of grams per year.

Now to plot the solution using Maple and to describe in words the effect of the variation in the incoming chemical,

we have: C(t) = 20 +ce ∧ -t/2, where c is an arbitrary constant since there was no chemical in the pond initially.

Therefore, C(t) = 0 , then C = -20

So that Q(t) = 20 ( 1 - e ∧ -t/2)

Kindly find the expression of the above statement C = C(t) below:

Note: In the diagram, Q is represented by C

Therefore, C = Q

Answer 2

Answer:

kindly find attachment for detailed answer

Explanation

Consider a pond that initially contains 10 million gallons of fresh water. Water containing a chemical pollutant flows into the pond at the rate of 5 million gallons per year (gal/yr), and the mixture in the pond flows out at the same rate. The concentration c=c(t) of the chemical in the incoming water varies periodically with time to the expression c(t) = 2 + sin(2t) grams per gallon (g/gal).

Construct a mathematical model of this flow process and determine the amount of chemical in the pond at any time t. Then, plot the solution using Maple and describe in words the effect of the variation in the incoming chemical.