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Chapter 3: Problem 16

Suppose a colony of bacteria has a continuous growth rate of \(70 \%\) per hour. How long does it take the colony to quadruple in size?

Short Answer

Expert verified
It takes approximately 2.02 hours for the colony of bacteria to quadruple in size.

Step by step solution

01

Understand exponential growth formula

The exponential growth formula is given by: \[P(t) = P_0 * e^{rt}\] where, \(P(t)\) = Population at time \(t\) \(P_0\) = Initial population \(r\) = Growth rate per unit of time \(t\) = Time In our case, we want to find the time taken for the colony to quadruple in size, so we'll have: \(P(t) = 4 * P_0\)
02

Substitute growth rate into the formula

The given continuous growth rate is 70% per hour, which can be converted to a decimal by dividing by 100: \[r = \frac{70}{100} = 0.7\] Now substitute the growth rate \(r = 0.7\) into the exponential growth formula: \[4 * P_0 = P_0 * e^{0.7t}\]
03

Solve for time 't'

To solve for the time 't', we can first divide both sides of the equation by \(P_0\): \[4 = e^{0.7t}\] Now, we need to take natural logarithm (represented as 'ln') of both sides of the equation, which will help us isolate the variable 't': \[\ln(4) = \ln(e^{0.7t})\] Using the property of logarithm, we can write: \[\ln(4) = 0.7t * \ln(e)\] Since the natural logarithm of \(e\) is 1, we get: \[\ln(4) = 0.7t\] Finally, solve for the time 't' by dividing both sides by 0.7: \[t = \frac{\ln(4)}{0.7}\]
04

Calculate the value of 't'

Now, just plug in the numbers into a calculator to find the value of 't': \[t \approx \frac{\ln(4)}{0.7} \approx 2.02\] So, it takes approximately 2.02 hours for the colony of bacteria to quadruple in size.

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