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# The leaves of a tree lose water to the atmosphere via the process of transpiration. A particular tree loses water at the rate of $3 \times 10^{-8} \mathrm{m}^{3} / \mathrm{s}$; this water is replenished by the upward flow of sap through vessels in the trunk. This tree's trunk contains about 2000 vessels, each $100 \mu \mathrm{m}$ in diameter. What is the speed of the sap flowing in the vessels?

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Given: $Q = 3 \cdot 10^{-8}$ m$^3$/s (total volume flow rate), $N = 2000$, $r_1 = 50 \cdot 10^{-6}$ m. The speed of the sap which flows in the vessels is:

\begin{align*} Q &= v \cdot A \\ v &= \frac{Q}{A} \\ &= \frac{Q}{N \cdot \pi r_1^2} \\ &= \frac{3\cdot 10^{-8}}{2000 \cdot 3.14 \cdot 2500 \cdot 10^{-12}} \\ &= 0.0019 \, \text{m/s.} \\ v &\approx 0.002 \, \text{m/s. }\end{align*}

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