- Topic
- Related rates
- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Related rates practice question 17
Doctors set the doses of some medicines by body surface area. A common estimate of a person’s body surface area, in square metres, is , where is height in centimetres and is mass in kilograms. A child is 144 cm tall with a mass of 36 kg, and is growing at 6 cm per year while gaining 4 kg per year. How fast is the child’s body surface area increasing?
Give the answer in square metres per year.
The child grows taller and heavier at the same time, so and both depend on . Which rules does the square root of the product call for?
Write and differentiate with respect to : the chain rule for the square root, and the product rule for .
. At this instant .
A rate in m² per year, exact or to three decimal places.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Surface area depends on height and mass, and both depend on time. Differentiate the formula with respect to t, using the chain rule for the square root and the product rule for hw, then substitute the values of this instant.
Working
Chain rule for the square root, product rule for hw.
The square root at this instant, so S = 72/60 = 1.2 m² now.
Growing taller and gaining mass both add surface.
Square metres per year.
Answer
The answer is 11/120 square metres per year, about 0.092.
Check
A year from now, at 150 cm and 40 kg, S = √6000/60 ≈ 1.291 m², about 0.091 m² more than the 1.2 m² of today.
Where this goes wrong
Differentiating hw as if only one factor changed. The child grows taller and heavier at once, so the derivative of hw is h′w + hw′; dropping either term gives 0.025 or about 0.067.
Leans on
The product rule, from Differentiation rules
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