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Chapter 3: Problem 7

Suppose a colony of bacteria has a continuous growth rate of \(15 \%\) per hour. By what percent will the colony have grown after eight hours?

Short Answer

Expert verified
The colony will have grown by approximately \(155.21\%\) after eight hours.

Step by step solution

01

Understand the Formula

The formula for continuous exponential growth is given by the equation \(P(t) = P_0(1 + r)^t\), where \(P_0\) is the initial population, \(r\) is the growth rate, \(t\) is the number of hours, and \(P(t)\) is the population after \(t\) hours.
02

Input Given Values

We are given the continuous growth rate of \(15\% = 0.15\), and the number of hours is \(8\). So, our formula becomes \(P(t) = P_0(1 + 0.15)^8\).
03

Calculate the Population Growth After Eight Hours

We have the formula \(P(t) = P_0(1 + 0.15)^8\). Let's calculate the population growth after eight hours. \(P(t) = P_0(1 + 0.15)^8 = P_0(1.15)^8\).
04

Evaluate (1 + Growth Rate)^Hours

Let's evaluate \((1.15)^8\): \((1.15)^8 \approx 2.55208\).
05

Calculate the Percent Growth

We have calculated that the ratio between the population after eight hours and the initial population is \(2.55208\). To convert this ratio into a percentage growth, we have to subtract \(1\) from the ratio and then multiply it by \(100\). So, we have \((2.55208 - 1) \times 100 \approx 155.21\%\). Hence, the colony will have grown by approximately \(155.21\%\) after eight hours.

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