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But water is still flowing into the lakes, so would it not eventually reach an equilibrium where water flows past the dam at the same rate it enters the lake?

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Evaporation is a huge problem for these lakes. By some estimates, each lake loses 800,000 acre feet of water to evaporation annually.

The water level is going down, so the rate leaving is already higher than the rate coming in.

If I understand it correctly, had there been no dam, the river would already be dry.

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