Question

Use the exponential growth model, A = A0 e^kt to show that the time is takes a population to double (to frow from A0 to 2 A0) is given by t = ln 2/k.

Answer 1

Answer:

Proof below

Step-by-step explanation:

Exponential Grow Model

The equation to model some time dependant event as an exponential is

$A=A_oe^{kt}$

Where Ao is the initial value, k is a constant and t is the time. With the value of Ao and k, we can compute the value of A for any time

We are required to find the time when the population being modeled doubles from Ao to 2 Ao. We need to solve the equation

$2A_o=A_oe^{kt}$

Simplifying by Ao

$2=e^{kt}$

Taking logarithms in both sides

$ln2=lne^{kt}$

By properties of logarithms and since lne=1

$ln2=kt\cdot lne=kt$

Solving for t

$\displaystyle t=\frac{ln2}{k}$

Hence proven