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Chapter 3: Problem 31

Suppose a colony of bacteria has tripled in five hours. What is the continuous growth rate of this colony of bacteria?

Short Answer

Expert verified
The continuous growth rate of the colony of bacteria is approximately 0.22, or 22% per hour.

Step by step solution

01

Write down the given values

We know that the initial population is \(P_0\), after 5 hours the population triples, which means the population becomes \(3 * P_0\). Moreover, the time elapsed, t, is 5 hours. Write the given information: - \(P_0 = P_0\) - \(P(t) = 3 * P_0\) - \(t = 5\) hours
02

Set up the exponential growth equation

Use the exponential growth formula: \( P(t) = P_0 * e^{rt} \) Substitute the given values into the equation: \( 3 * P_0 = P_0 * e^{5r} \)
03

Solve for the continuous growth rate (r)

Our goal is to find the growth rate (r). First, divide both sides of the equation by \(P_0\): \( 3 = e^{5r} \) Now, we need to isolate r. To do this, take the natural logarithm (ln) of both sides: \( ln(3) = ln(e^{5r}) \) Using the logarithm property \( ln(a^b) = b * ln(a) \), we get: \( ln(3) = 5r * ln(e) \) Since \( ln(e) = 1 \), the equation becomes: \( ln(3) = 5r \) Now, divide both sides by 5 to get r: \( r = \dfrac{ln(3)}{5} \)
04

Calculate the continuous growth rate

Plug in the value of ln(3) and divide by 5 to get the continuous growth rate: \( r = \dfrac{ln(3)}{5} \approx \dfrac{1.099}{5} \approx 0.22 \) The continuous growth rate of the colony of bacteria is approximately 0.22, or 22% per hour.

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