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Chapter 3: Problem 33

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 \(t=\frac{\ln 2}{k}\).

Short Answer

Expert verified
Therefore, the time it takes a population to double is given by \(t = \frac{\ln2}{k}\).

Step by step solution

01

Set up the equation

Begin by setting up the equation for exponential growth. We want the population to double, so the final amount \(A\) will be \(2A_{0}\). This gives us the equation \(2A_{0} = A_{0}e^{kt}\).
02

Simplify the equation

We can simplify this equation by dividing both sides by \(A_0\), which gives us: \(2 = e^{kt}\).
03

Apply the natural logarithm to both sides

Next, we need to solve for \(t\), which is in the exponent. To do that, we use the natural logarithm. Applying the logarithm to both sides of the equation gives us: \(\ln2 = \ln(e^{kt})\).
04

Utilize logarithmic properties

Our equation is still not solved for \(t\). To get closer, we can utilize the well-known logarithmic property that allows us to move the exponent as a multiple in front of the logarithm, i.e., \(\ln(e^{kt}) = kt * \ln e\). Since the natural logarithm of \(e\) is 1, this simplifies further to \(kt = \ln2\).
05

Solve for \(t\)

To finish our calculation, divide by \(k\) on both sides - we get \(t = \frac{\ln2}{k}\).

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Most popular questions from this chapter

Nigeria has a growth rate of 0.025 or \(2.5 \% .\) Describe what this means.

One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.

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