2. Metal Company is a mine for metals. Part of their process involves disposing of waste rock that contains a harmful substance that is water soluble. The waste is deposited into ponds that have clean water flowing into them and water flowing out at the same rate so that the volume of the pond stays the same. Let $x(t)$ be the mass of the harmful substance in the pond at time $t$ and $x(0) = x_0$ be the initial amount of the harmful substance in the pond at $t = 0$. Let $r_{in}$ and $r_{out}$ be the rate of water into and out of the pond. Let $V$ be the volume of the pond. (a) Assume for a moment that the pond does not have algae. Derive a differential equation to describe how the amount of the harmful substance in the pond changes with time. (b) Solve the differential equation derived in part (a). (c) A certain type of algae lives in the pond that degrades the harmful substance at a rate proportional to the square of the concentration of the harmful substance in the pond. Derive the differential equation that includes this new information using k as the proportionality constant for the degradation term.
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The rate at which the harmful substance enters the pond is given by the rate of water flowing into the pond (Tin) multiplied by the concentration of the harmful substance in the incoming water. The rate at which the harmful substance leaves the pond is given Show more…
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