Calculus Problem?

Question

The number of yeast cells in a laboratory culture increases rapidly at first but levels off eventually. The population is modeled by the function below, where t is measured in hours. At time t = 0 the population is 30 cells and is increasing at a rate of 21 cells/hour.

n = f(t) = a / 1+be^-0.8t

a) Find values for "a" and "b"
b) According to this model, in the long run the yeast population stabilizes at ______ cells.

Answer 1

We know:

f(0) = 30 = a/(1+b)

so

a = 30*(1+b)

We also know:

f'(0) = 21 = .8*a*b/(1+b)^2

so solving two equations in two unknowns gives
a = 240
b = 7

Now taking the limit as t goes to infinity, the denominator goes to one. Thus the population will stabilize at 240.

Answer 2

I assume you mean it is... a / ( 1 + b...etc )

for t = 0, n = 30

30 = a / ( 1 + b*e^0)
so 30 = a / ( 1 + b), or
30 +30 b = a,

or a = 30( 1 + b)


now second fact ---> dn / dt = 21
dn / dt = -a*[ (...)^-2]*( -0.8be^(-0.8t) )

and at t = 0, 21 = -[ a / ( 1 + b)^2 ]*[ -0.8 b]
or 21 = 0.8ab / ( 1 + b)^2


now solve by combining them for a and b...

21 = 0.8 b*(30*( 1 + b) ) / ( 1 + b )^2
21 = 24 b/ ( 1 + b), 21 + 21b = 24b, so 3b = 21, and b = 7

now a = 30 ( 1 + ( 7) ), so a = 240

as t ---> infinity, the value of n approaches
a / ( 1 + 0) , so n ---> 240

note...... [ as t --> inf, b*e^(-0.8t) will ---> 0 ], thus the ( 1 + 0 ) in the denom. of the step above....

Answer 3

for t=0
30= a/(1+b) so a= 30+30 b
at t=1
51=a/(1+b*e-0.8)

51+51e^-0.8 b = 30+30b so b=21/(30-51 e^-0.8) =2.6
and a=107.9
lim f(t) t==> infinity =a = 108 cells
If you take the derivate at t=0 you get 21= (24b+24b^2)/(1+b)^2
21+42b+21b^2) =24b+24b^2 3b^2-18b-21=0
b^2-6b-7=0 b=((6+-8))2 as a can´t be 0 b= 7 and a =240
which is the final population.
This seems to be the right interpretation
disregard the result of 108 cells

Answer 4

n(1+be^-0.8t) = a......(1)
30(1+b) = a
Differentiate (1) w.r.t time at t = 0,
-.8bn + n' (1+b) = 0 = > -.8b(30) + 21(1+b) = 0
Solve for b,
b = 7
a = 240
As t->infinity, we have
n = a = 240 cells