Problem 33 - Algebra and Trigonometry

Question

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(\left.2 A_{0}\right)\) is given by \(t=\frac{\ln 2}{k}\)

Step-by-Step Solution

Answer

The time (t) it takes a population to double in an exponential growth model is given by \(t = \frac{\ln2}{k}\). This is derived from the exponential growth equation by setting the final amount to twice the initial amount, simplifying, and then solving for t.

Step 1: Understand the Exponential Growth Model

An exponential growth model is represented by the equation \(A=A_0 e^{kt}\), where: \n- \(A\) is the final amount \n- \(A_0\) is the initial amount (i.e., the starting quantity) \n- \(e\) is the base of the natural logarithm (approx. 2.71828), showing that the quantity is growing exponentially \n- \(k\) is the growth rate (constant) \n- \(t\) is the time

Step 2: Apply the Doubling Condition

For a population to double, the final amount \(A\) will be twice the initial amount \(A_0\). Substituting \(2A_0\) for \(A\) in the growth equation results in the following equation: \(2A_0 = A_0 e^{kt}\)

Step 3: Simplify and Solve for t

Divide through the equation by \(A_0\) to get \(2=e^{kt}\). Next, take the natural logarithm (ln) of both sides to get \(\ln 2= kt\). Lastly, rearrange the equation to solve for \(t\), resulting in \(t = \frac{\ln 2}{k}\)