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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.

Respuesta :

elcharly64

Answer:

Proof below

Step-by-step explanation:

Exponential Grow Model

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

[tex]A=A_oe^{kt}[/tex]

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

[tex]2A_o=A_oe^{kt}[/tex]

Simplifying by Ao

[tex]2=e^{kt}[/tex]

Taking logarithms in both sides

[tex]ln2=lne^{kt}[/tex]

By properties of logarithms and since lne=1

[tex]ln2=kt\cdot lne=kt[/tex]

Solving for t

[tex]\displaystyle t=\frac{ln2}{k}[/tex]

Hence proven

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