Question

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.

Answer 1

Step-by-step explanation:

`n=f(t)= (a)/((1+be^(-.9t)))`

At t = 0

30 =
`(a)/(1 + b )`

30 + 30 b = a

`(dn)/(dt) =f(t)= (-.9abe^(-.9t))/((1+be^(-.9t))^2)`

For t = o

`(dn)/(dt) =f(t)= (.9ab)/((1+b)^2)`

given

24 =
`(.9ab)/((1+b)^2)`

24 =
`(30*.9b )/(1+b)`

24 = 27b / 1 + b

24 + 24 b = 27 b

24 = 3 b

b = 8

a = 30 + 30 x 8 = 270

`n=f(t)= (a)/((1+be^(-.9t)))`

Put t = infinity

n = a = 270

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