# Concentration and Differential Equations

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A polluted river with a nutrient concentration of 110g/m^3 is flowing at a rate of 130m^3/day into an estuary of volume 1000m^3. At the same time, water from the estruary is flowing into the ocean at 120m^3/day. The initial nutrient concentration in the estuary is 35g/m^3.

(i) Let N(t) be the amount of nutrient (in grams) in the estuary at time t. Write down and solve and solve an appropriate differential equation for N(t) along with the appropriate initial condition.

(ii) After a long time, what is the concentration of nutrient in the estuary?

(iii) It is known that if the nutrient concentration in the estuary reaches 85g/m^3 an algal bloom will occur. How many days does it take for the nutrient concentration to reach this threshold?

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Solution:

Volume of estuary = V = 1000 m3

Initial nutrient concentration in estuary water = ne0 = 35 g/m3

Initial mass of nutrients in the estuary = Initial concentration x Volume of estuary = 35 x 1000 = 35000 g

Nutrient concentration in river water = n1 = 110 g/m3

Rate of flow of river water into the estuary Rre = 120 m3/day = 120/(24x60x60) m3/sec or 1.39x10-3 m3/sec

Rate of flow of estuary water into the ocean Reo = 120 m3/day = 1.39x10-3 m3/sec

i) Let us consider a time instant t at which total mass of nutrients in the estuary be N(t). Then, nutrient concentration in the estuary at time t = Mass of nutrients at time t/Volume of estuary = N/1000 g/m3

Let us ...

#### Solution Summary

This solution helps with questions regarding concentrations and differential equations.