- Topic
- Related rates
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Related rates practice question 13
Poiseuille’s law gives the rate of blood flow through a vessel of radius as , where is the pressure difference along the vessel and is a constant. As an artery narrows, its radius falls at an instantaneous rate of 1 % per minute while the pressure difference rises at 2 % per minute; that is, and . Find the percentage rate of change of the blood flow .
Give a signed percentage per minute.
The answer is a percentage rate, , and the radius enters to the fourth power. What does taking logarithms do to a product, and to a power?
Percentage rates are logarithmic derivatives, . Take logarithms: .
Differentiate with respect to : . Keep the sign of .
A number, the percentage, negative if the flow is falling.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Percentage rates of change are logarithmic derivatives. Taking log of the law turns the product into a sum and the fourth power into a factor of 4, and differentiating gives the percentage rates directly.
Working
Logarithm of a product, and log(r⁴) = 4 log r.
Differentiate with respect to t; log k is a constant.
The flow falls by 2 % per minute: a 1 % narrowing outweighs a 2 % rise in pressure, because the radius enters to the fourth power.
Answer
The answer is the flow falls at 2 per cent per minute, a rate of −2 % per minute.
Check
In a minute r falls to about 0.99 of itself and P rises to about 1.02 of itself, so F becomes about 1.02 × 0.99⁴ ≈ 0.980 of itself, 2 % less.
Where this goes wrong
Forgetting the fourth power, which gives 2 − 1 = +1 %. Since log(r⁴) = 4 log r, the radius changes the flow four times as fast as it changes itself, in percentage terms.
Leans on
Logarithmic differentiation, from The chain rule and logarithms
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