use the exponential growth model, ( a = a_0 e^{kt} ), to show that the time it takes a population to double…

use the exponential growth model, ( a = a_0 e^{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_0 e^{kt} ) substitute.\n( 2 = e^{kt} ) divide both sides by ( a_0 ).\n( ln 2=ln square ) take the natural logarithm on both sides.

use the exponential growth model, ( a = a_0 e^{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_0 e^{kt} ) substitute.\n( 2 = e^{kt} ) divide both sides by ( a_0 ).\n( ln 2=ln square ) take the natural logarithm on both sides.

Answer

Explanation:

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

Given the exponential - growth model (A=A_0e^{kt}), when (A = 2A_0), we have (2A_0=A_0e^{kt}).

Step2: Simplify the equation

Divide both sides of the equation (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), if (y = e^{x}), then (\ln(y)=x). So, (\ln(2)=\ln(e^{kt})). Since (\ln(e^{kt}) = kt) (by the property (\ln(e^{u})=u) where (u = kt)), we have (kt=\ln(2)).

Step4: Solve for (t)

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

Answer:

The time (t) it takes for the population to double is (t = \frac{\ln(2)}{k})