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Chapter 3: Problem 34

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to triple (to grow from \(A_{0}\) to \(3 A_{0}\) ) is given by \(t=\frac{\ln 3}{k}\).

Short Answer

Expert verified
The time it takes for a population to triple under exponential growth is given by \(t=\frac{\ln 3}{k}\).

Step by step solution

01

Identify The Given Variables

Here, we are given \(A=A_{0} e^{k t}\) where \(A\) is final population, \(A_{0}\) is initial population, \(k\) is growth rate and \(t\) is time. We know the final population is three times the initial population, so we sub in \(3 A_{0}\) for \(A.\)
02

Substitute Given Variables In The Formula

Substitute \(3 A_{0}\) for \(A\) into the equation. This gives us the equation \(3 A_{0}=A_{0} e^{k t}\). Since we want to solve for \(t\) and isolate it on one side of the equation, we should get rid of \(A_{0}\) on both sides.
03

Simplify The Equation

Both sides of the equation have \(A_{0}\), so we can divide both sides by \(A_{0}\), which would give us: \(3 = e^{k t}\)
04

Take The Natural Logarithm On Both Sides

To isolate \(k t\), we need to take the natural logarithm on both sides. This gives: \(\ln 3 = \ln e^{k t}\). Now, the right side can be simplified using the logarithmic property \(\ln e^{x}=x\), which would make it \(k t\). Hence, we get: \(\ln 3 = k t\)
05

Solve For \(t.\)

Finally, divide both sides by \(k\) to isolate the \(t\) term alone on one side. This gives us the final formula: \(t=\frac{\ln 3}{k}\)

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