Question

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−0.7t where t is measured in hours. At time t = 0 the population is 30 cells and is increasing at a rate of 18 cells/hour. Find the values of a and b.

Answer 1

Answer:

a = 30

b = 6/7

Step-by-step explanation:

The number of yeast cells after t hours is modeled by the following equation:

$f(t) = a(1 + be^{-0.7t})$

In which a is the initial number of cells.

At time t = 0 the population is 30 cells

This means that $a = 30$

So

$f(t) = 30(1 + be^{-0.7t})$

And increasing at a rate of 18 cells/hour.

This means that f'(0) = 18.

We use this to find b.

$f(t) = 30(1 + be^{-0.7t})$

So

$f(t) = 30 + 30be^{-0.7t}$

Then, it's derivative is:

$f'(t) = -30*0.7be^{-0.7t}$

We have that:

f'(0) = 18

So

$f'(0) = -30*0.7be^{-0.7*0} = -21b$

Then

$-21b = 18$

$21b = -18$

$b = -\frac{18}{21}$

$b = \frac{6}{7}$