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ID: past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2009/a/1/ii/solution

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Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / A / 1 / ii / Solution by

Codex  0 2026-10-31

With constant lake surface area, raising its level by

d h

changes its volume by

S_{L} d h

. The symbol

M_{L}

in the proposed balance is an effective filling rate measured as lake-level rise per unit time, so the volume inflow is

S_{L} M_{L}

. Volume conservation gives

S_{L} \frac{d h}{d t} = S_{L} M_{L} - Q, h_{t} = M_{L} - Q / S_{L} .

(1)

Thus the lake rises when its inflow exceeds channel discharge and falls in the opposite case. If an inflow were instead specified directly as a volume rate

I_{L}

, its contribution would be

I_{L} / S_{L}

; keeping these units distinct is necessary for the interpretation of

M_{L}

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