Question

HELP NEEDED:
The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function
n = f(t) = a/(1 + be−0.9^t)
where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 16 cells/hour. Find the values of a and b.

Answer 1

you need two equations to solve for two unknowns a and b

Solve for f(0) and f(1) then find a and b with this system of equations

Answer 2

Hey, I cant seem to find the two equations

Answer 3

When t=0, n=20 and when t=1, n=36.

Answer 4

Plug in values to make 2 equations and solve the system to find a and b.

Answer 5

where is t1, n=36 coming from

Answer 6

Since, the population is increasing at 16 cells per hr.

Answer 7

We were given that n=20 at t=0, so at t=1, n=20+16.

Answer 8

We could also say that when t=3, n=20+16+16 but since we only have 2 constants to fine, we only need 2 equations.

Answer 9

I see the logic but do you get the same answer if you get two equations
f(0) = 20 and f'(0) = 16 since rate of change is first derivative

Answer 10

I think that should work too.

Answer 11

Is the function n=a/(1+be^[-0.9t])?

Answer 12

that looks correct

Answer 13

I got a=79.66 and b=2.98

Answer 14

Using the system of equations method.

Answer 15

is it the right answer?

Answer 16

I am working it again with the derivative method to check it.

Answer 17

20 = a/(1+b)
16 = 9ab/(1+b)^2

Answer 18

.9

Answer 19

Yes, those are the equations I got using the derivative method.

Answer 20

Solving those equations simultaneously, I got a=180, b=8. Let me check my work. Just a minute.

Answer 21

Hi guys, how did the second equation ?

Answer 22

20 = a/(1+b)
16 = .9ab/(1+b)^2

Answer 23

f'(0)

Answer 24

can you show me?

Answer 25

Neither answer that I got is a function that starts at 20, increases rapidly, then levels off. I think I may have misunderstood the original equation.

Answer 26

To get the second equation, you use the quotient rule and plug in 0 for t.

Answer 27

@bobobobobb , do you know how to calculate a derivative using the quotient rule?

Answer 28

Yes, are you just diffrentiating a/1+b?

Answer 29

did you use quadratic formula to find b

Answer 30

No, I differentiated the original equation.

Answer 31

when you take the first derivative plug in t =0 you end up with the second equation
if you make a the subject in the first equation and plug it in in the second equation, you end with a quadratic equation to solve for b