- Topic
- Related rates
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Related rates practice question 5
A salt crystal grows in the shape of a cube, and each of its edges lengthens at 0.5 mm per day. How fast is the total surface area of the crystal increasing at the moment each edge is 5 mm long?
Give the answer in square millimetres per day.
The question is about surface area, not volume. How does the total area of a cube’s faces depend on its edge , and when should go in?
A cube with edge has six square faces, so its surface area is . Differentiate with respect to ; the chain rule brings in .
A rate in mm² per day, exact or to two decimal places.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Write the surface area in terms of the edge, differentiate with respect to time while the edge is still a variable, and only then substitute the edge and its rate at this instant.
Working
Chain rule: the edge a depends on t.
Substitute a = 5 and da/dt = 0.5 only now.
Answer
The answer is 30 square millimetres per day.
Check
In 0.1 day the edge reaches 5.05 mm and the surface area goes from 150 to about 153.0 mm², a rate of about 30 mm²/day.
Where this goes wrong
Substituting a = 5 into S = 6a² first, getting the constant 150, and differentiating that to get 0. The edge is a variable until the differentiation is done.
Leans on
The chain rule, from The chain rule and logarithms
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