- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Exponential growth and decay practice question 5
A yeast culture grows at a rate proportional to its size, and increases by 50 % every 4 hours. What is its doubling time?
Give the answer in hours.
Growth proportional to size multiplies the culture by the same factor over every 4 hours. What is that factor, and what factor does doubling ask for?
With , increasing by 50 % in 4 hours means , so .
Doubling means . Solve for 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
Proportional growth is exponential. One fact fixes the growth constant; the doubling time then follows from ekt = 2.
Working
The 50 % increase over 4 hours.
Doubling time. The exact form is the answer; the decimal is only a sense of scale.
Answer
The answer is 4 log 2 over log 1.5 hours, about 6.84 hours.
Check
6.84 hours is 1.71 periods of 4 hours, and 1.51.71 ≈ 2.00.
Where this goes wrong
Doubling the 4 hours to get 8, on the grounds that two 50 % increases make 100 %. Two 50 % increases multiply to 2.25, not 2; the doubling happens earlier, after about 6.84 hours.
Leans on
Laws of logarithms, from Functions, graphs and inverses
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