I have answered the first two questions, please answer the last three, thankyou!

Question

I have answered the first two questions, please answer the last three, thankyou!

A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration C = C(t) of the glucose solution in the bloodstream is

dC/dt = r ? k C where k is a positive constant.

(1) Suppose that the concentration at time t = 0 is C0. Determine the concentration at any time t by solving the differential equation.
C(t) = (to enter C0 in your answer, type "C_0", that is, "capital C, underscore, zero")
Answer: (Co - r/k)e^(-kt)+r/k

(2) Find lim t?? C(t) =

Answer: r/k

??=as t approaches infinity

(3) If C0 > lim t ->oo C(t), then (choose one):

(a)C(t) increases up toward its limit lim t?? C(t) as t ? ?.

(b)C(t) stays constant and is equal to its limit lim t?? C(t) as t ? ?.

(c)C(t) decreases down toward its limit lim t?? C(t) as t ? ?.

(4) If C0 = lim t?? C(t), then (choose one):

(a)C(t) increases up toward its limit lim t?? C(t) as t ? ?.

(b)C(t) stays constant and is equal to its limit lim t?? C(t) as t ? ?.

(c)C(t) decreases down toward its limit lim t?? C(t) as t ? ?.

(5) If C0 < lim t?? C(t), then (choose one):

(a)C(t) increases up toward its limit lim t?? C(t) as t ? ?.

(b)C(t) stays constant and is equal to its limit lim t?? C(t) as t ? ?.

(c)C(t) decreases down toward its limit lim t?? C(t) as t ? ?.

Explanation / Answer

3)

If C0 > lim t -> infinity C(t), then (choose one):

C(t) = (Co - r/k)e^(-kt)+ r/k

Now, given that C0 > lim t --> inf C(t)

So, C0 > r/k

And this means, C(t) must reduce so as to make C0 = r/k

So, option C

4) If C0 = lim t --> inf C(t), then (choose one):

Since C0 = r/k, it stays constant

So, option B

5) If C0 < lim t --> inf C(t), then (choose one):

Since C0 < r/k, it needs to increase

So, option A

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The following examples will help in understanding it :

For #3 :
Say C(t) = 40e^(-t) + 20
In this case, C0 = 60 --> so it starts with 60
And limit t --> inf = 20
So, it starts at 60 and then reduces to 20
Therefore, constituting a DECREASE
So, for 3, it is a decrease.

For #4 :
Say C(t) = Ae^(-t) + C
In this case, at t = 0 :
C(0) = C0 = A + C
And as t --> inf, C(t) = C
Also given that A + C = C --> A = 0
So, this means, the equation is infact C(t) = 0e^(-t) + C
C(t) = C ---> constant equation
So, for 4, it is a constant

For #5 :
Say C(t) = Ae^(-t) + C
C0 = A + C
As t --> inf, concentration = C
Given A + C < C
So, A < 0
As an example, lets take L
C(t) = -10e^(-t) + 30
C0 = -10 + 30 = 20
As t --> inf, C(t) = 30
So, C0 < the lim t --> inf C(t)
This shows that since it starts at 20 and as t --> inf, goes to 30
Therefore, an INCREASE
so, #5 is an increase

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