use the exponential growth model, ( a = a_0e^{kt} ), to show that the time it takes a population to double…

use the exponential growth model, ( a = a_0e^{kt} ), to show that the time it takes a population to double (to grow from ( a_0 ) to ( 2a_0 )) is given by ( t=\frac{ln 2}{k} ).\nto find the time it takes to grow the population from ( a_0 ) to ( 2a_0 ), substitute ( a = 2a_0 ).\n( 2a_0=a_0e^{kt} ) substitute.\n( square = e^{kt} ) divide both sides by ( a_0 ).

use the exponential growth model, ( a = a_0e^{kt} ), to show that the time it takes a population to double (to grow from ( a_0 ) to ( 2a_0 )) is given by ( t=\frac{ln 2}{k} ).\nto find the time it takes to grow the population from ( a_0 ) to ( 2a_0 ), substitute ( a = 2a_0 ).\n( 2a_0=a_0e^{kt} ) substitute.\n( square = e^{kt} ) divide both sides by ( a_0 ).

Answer

Explanation:

Step1: Substitute (A = 2A_0) into the formula

Given (A=A_0e^{kt}), when (A = 2A_0), we have (2A_0=A_0e^{kt}).

Step2: Simplify the equation

Divide both sides of (2A_0 = A_0e^{kt}) by (A_0) (since (A_0\neq0)). We get (2=e^{kt}).

Step3: Take the natural logarithm of both sides

Take the natural - logarithm of both sides of the equation (2 = e^{kt}). Using the property (\ln(e^{x})=x), we have (\ln(2)=\ln(e^{kt})). So, (\ln(2)=kt).

Step4: Solve for (t)

Divide both sides of (\ln(2)=kt) by (k) (assuming (k\neq0)). We get (t = \frac{\ln(2)}{k}).

Answer:

The value in the box is (2).