/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Suppose a colony of bacteria has... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 30

Suppose a colony of bacteria has doubled in two hours. What is the approximate continuous growth rate of this colony of bacteria?

Short Answer

Expert verified
The continuous growth rate for the colony of bacteria is approximately \(34.66\%\) per hour. We found this by using the continuous compound growth formula, \(P(t) = P_0e^{rt}\), and solving for \(r\) with the given information that the population doubles in two hours.

Step by step solution

01

Recall the formula for continuous compound growth

The formula for continuous compound growth is: \(P(t) = P_0e^{rt}\) where, - \(P(t)\) is the population at time \(t\) - \(P_0\) is the initial population - \(r\) is the continuous growth rate - \(e\) is the base of the natural logarithms (approximately equal to 2.718) - \(t\) is the time In our case, we know that the population doubles in two hours, so we can write the equation as: \(2P_0 = P_0e^{2r}\) We need to find the value of \(r\).
02

Solve for the continuous growth rate

To solve for \(r\), we will follow these steps: 1. Divide both sides of the equation by \(P_0\): \(\frac{2P_0}{P_0} = \frac{P_0e^{2r}}{P_0}\) which simplifies to: \(2 = e^{2r}\) 2. Take the natural logarithm of both sides of the equation \(ln(2) = ln\left(e^{2r}\right)\) Using the property that \(ln(a^b) = b\ln(a)\), we can rewrite the equation as: \(ln(2) = 2r\cdot ln(e)\) Since \(ln(e) = 1\), the equation simplifies to: \(ln(2) = 2r\) 3. Divide both sides of the equation by 2: \(\frac{ln(2)}{2} = r\)
03

Compute the approximate continuous growth rate

Now, we compute the continuous growth rate by dividing the natural logarithm of 2 by 2: \(r \approx \frac{ln(2)}{2}\) Using a calculator, we get approximately: \(r \approx 0.3466\)
04

Express the growth rate as a percentage

To convert the growth rate to a percentage, multiply the growth rate by 100: \(r_\% \approx 0.3466 \times 100\) \(r_\% \approx 34.66\%\) The continuous growth rate for the colony of bacteria is approximately 34.66% per hour.

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