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Chapter 3: Problem 32

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

Short Answer

Expert verified
The continuous growth rate of the colony of bacteria is approximately 0.5493 per hour.

Step by step solution

01

Write down the formula and given values

Write down the given values and the exponential growth formula: Final Population = 3 脳 Initial Population time = 2 hours Exponential growth formula: Final Population = Initial Population 脳 e^(growth rate 脳 time)
02

Substitute the known values into the formula

Substitute the known values (Final Population and time) into the growth formula: \(3 \times Initial Population = Initial Population \times e^(growth rate \times 2)\)
03

Simplify the equation

Simplify the equation by dividing both sides by Initial Population: \(3 = e^(growth\_rate \times 2)\)
04

Apply natural logarithm to both sides of the equation

Apply the natural logarithm (ln) to both sides of the equation to isolate the exponential term: \(ln(3) = ln(e^(growth\_rate \times 2))\)
05

Use the logarithmic property

Use the logarithmic property \(ln(a^b) = b \times ln(a)\) to simplify the equation further: \(ln(3) = 2 \times growth\_rate \times ln(e)\) Since \(ln(e) = 1\), the equation can be further simplified: \(ln(3) = 2 \times growth\_rate\)
06

Solve for the continuous growth rate

Solve for the continuous growth rate by dividing both sides by 2: \(growth\_rate = \frac{ln(3)}{2}\) Now, calculate the value of the growth rate: \(growth\_rate 鈮� \frac{1.0986}{2} 鈮� 0.5493\)
07

Conclusion

The continuous growth rate of the colony of bacteria is approximately 0.5493 per hour.

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