2. A small lake of volume 100 cubic meters has accidentally been polluted by 10 000 kilograms of a highly toxic substance. A river flows into and out of the lake at a rate of 20 000 cubic meters per hour. Assuming that the entering river contains fresh water and that the toxic substance is fully mixed throughout the lake at all times. find the number of hours for the mass of the pollutant to decrease to an acceptable 100 kilograms.
To solve this problem, we have to use the concept of differential equations and modeling the rate of change of pollutant concentration in the lake over time. Let's denote:
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We want to find the time it takes for the mass of the pollutant (
The change in mass of the pollutant in the lake over time is given by the rate of inflow of pollutant minus the rate of outflow of pollutant:
Where:
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We also know that
Now, we can substitute the given values into the equation:
Simplify the equation:
Now, we have a first-order differential equation. We can solve it by separating variables:
Integrate both sides:
Where
Now, we can solve for
Since
Now, we can solve for
Simplify:
Now, we can find the time
Step by step
Solved in 3 steps