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Question: 1 10 pts suppose we have a newlybuilt factory that...

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1. (10 pts) Suppose we have a newly-built factory that sits on a small lake with a volume of 30 cubic kilometers Your friend comes to you, concerned about the amount of pollution being dumped into the lake. They have carefully constructed a plan to clean the lake, and would like your opinion on it. The factory causing the pollution cannot be completely stopped from dumping pollutant, but your friend wants to begin to filter the lake water to remove pollution. Having collected some data from their lake, their plan is centered on three observations . The pollutant from the factory enters the lake at a rate of sin (t) +1 cubic kilometers, where t denotes the . There was initially no pollution in the lake, meaning that if p(0) denotes the amount of pollution in the . There are enough volunteers to filter exactly half of the lake water every day. Filtering lake water will Assuming that the pollutant is perfectly mixed into the lake, use Chapter 2 methods to prepare a recommendation » A differential equation modeling p(t), the amount of pollutant in the lake at time t under your friends plan number of days after t 0, the day the factory opened. lake at time t, then p(0)0 remove all pollutant from it for your friend regarding if they should proceed or not. Be sure to include the following: An explicit solution to your given differential equation. . A graph of your solution to the proposed IVP using dfield. A discussion of the long term behavior of the solution (what happens to p as too? Based on this, is your friends plan effective?) (Hint: Your differential equation should be first order linear. You will need to integrate the product of an exponential function and a trig function, which requires an integration by parts trick. If you dont remember how the trick works, consult your Calc 2 textbook on integration by parts)

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