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Chapter 3: Problem 8

Suppose a colony of bacteria has a continuous growth rate of \(20 \%\) per hour. By what percent will the colony have grown after seven hours?

Short Answer

Expert verified
After seven hours, the bacterial colony will have grown by approximately 305.5%.

Step by step solution

01

Convert growth rate to decimal form

First, we need to convert the 20% growth rate into decimal form, which is done by dividing by 100. So, the growth rate in decimal form: \[ r = \frac{20}{100} = 0.2 \]
02

Set up the exponential growth formula

We have seven hours as the time, and the growth rate is 0.2. We can now set up the continuous exponential growth formula: \[ P(t) = P_0 * e^{rt} \] where \(t = 7\) hours and \(r = 0.20\).
03

Solve for P(t)

Now, we want to find the ratio of the final population to the initial population. To cancel out the initial population, we simply divide both sides by \(P_0\): \[ \frac{P(t)}{P_0} = e^{rt} \] Plugging in our values of \(r = 0.20\) and \(t = 7\) hours, we get: \[ \frac{P(t)}{P_0} = e^{0.2 * 7} \]
04

Evaluate the expression

Calculate the expression on the right side of the equation: \[ \frac{P(t)}{P_0} = e^{1.4} \approx 4.055 \]
05

Calculate the growth percentage

To find the growth percentage, we can use the following formula: \[ Growth \, Percentage = (\frac{P(t)}{P_0} - 1) * 100\% \] Plugging in our final ratio of approximately 4.055, we have: \[ Growth \, Percentage 鈮� (4.055 - 1) * 100\% 鈮� 3.055 * 100\% 鈮� 305.5\% \] After seven hours, the bacterial colony will have grown by approximately 305.5%.

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