- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Exponential growth and decay practice question 3
A bacterial culture has population million after hours. How long does it take the population to double?
Give the answer in hours.
Doubling means reaching twice the starting population. What is the population at , and which tool brings down from an exponent?
Doubling means . Set and solve with a logarithm.
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §11.4CLP-1 §3.3Key results
Worked solution
Strategy
The doubling time of an exponential does not depend on the starting amount; solving ekt = 2 gives it directly.
Working
The 5 cancels: doubling time is independent of the initial population.
Take logarithms.
Answer
The answer is 5 natural log 2 hours, about 3.47.
Check
At t = 3.47 the population is 5e0.694 ≈ 10.0 million, twice the 5 million at the start.
Where this goes wrong
Solving 0.2t = 2 to get t = 10, forgetting the exponential. The equation to solve is e0.2t = 2, and a logarithm is needed to bring t down.
Leans on
Solving an exponential equation with logarithms, from Functions, graphs and inverses
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