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A growing crystal

CalculusMedium

A cube-shaped crystal grows so that each edge lengthens at a constant $0.2$ mm per hour. How fast is its volume increasing, in mm³ per hour, at the moment the edge is $5$ mm long? Give the answer as an integer.

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Hint

Write the volume as a function of the edge and differentiate with respect to time.

Solution

With edge $s$ , the volume is $V = s^{3}$ .
Differentiating in time gives $\frac{d V}{d t} = 3 s^{2} \times \frac{d s}{d t}$ .
At $s = 5$ and $\frac{d s}{d t} = 0.2$ , this is $3 \times 25 \times 0.2 = 15$ .
So the volume grows at $15$ mm³ per hour.

Answer: $15$

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