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Chapter 3: Problem 15

A colony of bacteria is growing exponentially, doubling in size every 100 minutes. How many minutes will it take for the colony of bacteria to triple in size?

Short Answer

Expert verified
It will take approximately 158.5 minutes for the colony of bacteria to triple in size.

Step by step solution

01

Define the exponential growth formula

The exponential growth formula is given by the equation \(A = A_0 \cdot 2^{\frac{t}{t_{double}}}\), where \(A\) is the final amount of bacteria, \(A_0\) is the initial amount of bacteria, \(t\) is the time in minutes, and \(t_{double}\) is the time it takes for the amount of bacteria to double.
02

Determine the time it takes for the colony to triple in size

We want to find out when the colony triples in size, therefore, we can rewrite the exponential growth formula as: \(3A_0 = A_0 \cdot 2^{\frac{t}{t_{double}}}\). Now, we need to solve for \(t\), when the remaining variables are given.
03

Simplify the equation

Divide both sides of the equation by \(A_0\) to eliminate it from the equation. This results in: \(3 = 2^{\frac{t}{t_{double}}}\).
04

Use logarithm to solve for t

To solve for \(t\), we can take the logarithm base 2 of both sides of the equation: \(\log_{2}(3) = \log_{2}(2^{\frac{t}{t_{double}}})\). Using the logarithm property, this equation can be further simplified to: \(\log_{2}(3) = \frac{t}{t_{double}}\).
05

Isolate t

Now, multiply both sides by \(t_{double}\) to isolate \(t\): \(t = t_{double} \cdot \log_{2}(3)\). In the problem, we're given that \(t_{double} = 100\), so now we need to plug that into the equation and solve for \(t\).
06

Calculate the time for the colony to triple in size

Plugging the doubling time into the equation, we have: \(t = 100 \cdot \log_{2}(3)\). Now, using a calculator to find the value of \(\log_{2}(3) \approx 1.585\), we can then calculate \(t\): \(t \approx 100 \cdot 1.585 = 158.5\)
07

Conclusion

Therefore, it will take approximately 158.5 minutes for the colony of bacteria to triple in size.

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