- Topic
- Related rates
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Related rates practice question 3
Three quantities depend on time and are related by . At a certain moment is increasing at an instantaneous rate of 6 % per year and is increasing at 2 % per year; that is, and . Find the percentage rate of change of at that moment.
Give the answer as a percentage per year.
Every rate here is a percentage rate, of the form . What is the derivative of with respect to , and what does a logarithm do to a quotient?
Percentage rates are . Take logarithms of the relation, , and differentiate.
A number.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Percentage rates of change are logarithmic derivatives. Taking log of a quotient turns it into a difference, and differentiating gives the percentage rates directly.
Working
Logarithm of a quotient.
Differentiate with respect to t; each term is a percentage rate.
Answer
The answer is R increases at 4 per cent per year.
Check
Taking P = e0.06t and Q = e0.02t makes R = e0.04t, which grows at exactly 4 % per year.
Where this goes wrong
Dividing the percentages, 6/2 = 3, because R is a quotient. Percentage rates of a quotient subtract; they do not divide.
Leans on
Laws of logarithms, from Functions, graphs and inverses
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