The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function а n = f(t) = 1 + be-0.90 where t is measured in hours. At time t = 0 the population is 30 cells and is increasing at a rate of 24 cells/hour. Find the values of a and b. a = b = According to this model, what happens to the yeast population in the long run? O The yeast population will shrink to 0 cells. The yeast population will stabilize at 270 cells. The yeast population will stabilize at 135 cells. The yeast population will stabilize at 8 cells. O The yeast population will grow without bound.

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Xema8 Xema8 · Answer 1 · 2020-10-17T09:16:29+02:00

Answer:

Explanation:

[tex]n=f(t)= \frac{a}{(1+be^{-.9t})}[/tex]

At t = 0

30 = [tex]\frac{a}{1 + b }[/tex]

30 + 30 b = a

[tex]\frac{dn}{dt} =f(t)= \frac{-.9abe^{-.9t}}{(1+be^{-.9t})^2}[/tex]

For t = o

[tex]\frac{dn}{dt} =f(t)= \frac{.9ab}{(1+b)^2}[/tex]

given

24 = [tex]\frac{.9ab}{(1+b)^2}[/tex]

24 = [tex]\frac{30\times.9b }{1+b}[/tex]

24 = 27b / 1 + b

24 + 24 b = 27 b

24 = 3 b

b = 8

a = 30 + 30 x 8 = 270

[tex]n=f(t)= \frac{a}{(1+be^{-.9t})}[/tex]

Put t = infinity

n = a = 270

So at infinite time yeast population will stabilise at number 270 .