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Chapter 4: Problem 15

Suppose a colony of bacteria has a continuous growth rate of \(35 \%\) per hour. How long does it take the colony to triple in size?

Short Answer

Expert verified
The bacteria colony will take approximately \(t = \frac{\ln{3}}{0.35} \approx 3.16\) hours to triple in size.

Step by step solution

01

Determine the Exponential Growth Formula

The general formula for exponential growth is given by: \[P(t) = P_0e^{rt}\] Where: - \(P(t)\) is the population at time t - \(P_0\) is the initial population - \(r\) is the growth rate (expressed as a decimal) - \(t\) is the time (in hours, in this case)
02

Set up the equation with given information

Since the bacteria colony triples in size, we can represent the final population as 3\(P_0\). The continuous growth rate is 35% per hour, so the growth rate as a decimal is 0.35. Using this information, we will set the exponential growth formula like this: \[3P_0 = P_0e^{0.35t}\]
03

Solve for the variable t

To find the time it takes for the bacteria colony to triple, we need to solve for t. First, we need to cancel out the \(P_0\) on both sides of the equation: \[3 = e^{0.35t}\] Now take the natural logarithm (ln) of both sides of the equation: \[\ln{3} = \ln{e^{0.35t}}\] Apply the power rule for logarithms on the right side, which states that \(\ln{a^b}=b\ln{a}\): \[\ln{3} = 0.35t\ln{e}\] Since \(\ln{e}=1\): \[\ln{3} = 0.35t\] Finally, isolate the variable \(t\): \[t = \frac{\ln{3}}{0.35}\]
04

Calculate the time

Plug in the values and calculate the value of t: \[t = \frac{\ln{3}}{0.35} \approx 3.16\] The bacteria colony will take approximately 3.16 hours to triple in size.

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