- Topic
- Related rates
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Related rates practice question 2
A circular ripple on a pond expands so that its radius grows at a constant 2 cm/s. How fast is the area inside the ripple increasing at the instant the radius is 4 cm?
Give the answer in square centimetres per second.
Which formula ties the area inside the ripple to its radius at every instant? Keep the radius a variable until after you differentiate.
Relate area to radius, , then differentiate both sides with respect to time.
An exact expression, or a decimal.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Write the relation that holds at every instant, differentiate it with respect to t, and only then substitute the given instant.
Working
Chain rule: r depends on t.
Substitute r = 4 and dr/dt = 2.
Answer
The answer is 16 pi square centimetres per second, about 50.3.
Check
Over the next 0.01 s the radius grows to 4.02 cm and the area by about 0.504 cm², a rate of about 50.4 cm²/s.
Where this goes wrong
Substituting r = 4 into A = πr² first, getting A = 16π, and then differentiating a constant to get 0. The radius is a variable until the very last step.
Leans on
Area of a circle, from Functions, graphs and inverses
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