Question

use the exponential growth model, ( a = a_0 e^{kt} ), to show that the time ( t ) (to grow from ( a_0 ) to ( 2a_0 )) is given by ( t = \frac{ln 2}{k} ).

to find the time it takes to grow the population from ( a_0 ) to ( 2a_0 ), substitute ( a = 2a_0 ).

substitute.

( square = a_0 e^{kt} )

Explanation:

Step1: Recall Exponential Growth Model

The exponential growth model is given by \( A = A_0 e^{kt} \), where \( A \) is the final amount, \( A_0 \) is the initial amount, \( k \) is the growth rate, and \( t \) is time.

Step2: Substitute \( A = 2A_0 \)

We want to find the time \( t \) when the population doubles, so we substitute \( A = 2A_0 \) into the equation \( A = A_0 e^{kt} \). This gives us \( 2A_0 = A_0 e^{kt} \).

Step3: Divide Both Sides by \( A_0 \)

Since \( A_0
eq 0 \) (as it represents an initial population), we can divide both sides of the equation \( 2A_0 = A_0 e^{kt} \) by \( A_0 \). This simplifies to \( 2 = e^{kt} \).

Step4: Take Natural Logarithm of Both Sides

To solve for \( t \), we take the natural logarithm of both sides of the equation \( 2 = e^{kt} \). Using the property \( \ln(e^x)=x \), we get \( \ln(2)=\ln(e^{kt}) \), which simplifies to \( \ln(2)=kt \).

Step5: Solve for \( t \)

We divide both sides of the equation \( \ln(2)=kt \) by \( k \) (assuming \( k
eq0 \)) to isolate \( t \). This gives us \( t = \frac{\ln(2)}{k} \), which is the formula for the doubling time.

Answer:

The time it takes for a population to double is given by \( t=\frac{\ln(2)}{k} \), and the substitution step gives \( 2A_0 = A_0 e^{kt} \) (when substituting \( A = 2A_0 \) into \( A = A_0 e^{kt} \)).