TestReview - This is a broad overview of the material that...

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This is a broad overview of the material that has been covered this semester. It is not a formal review but will provide good practice for the exam. Know how to at least do all of the following problems. Solve following differential equation: 2.2:1) 2.2:2) 2.2:3) 2.3:2) A tank initially contains 120L of pure water. A mixture containing a concentration of γ g/L of salt enters the tank at a rate of 2 L/min, and the well-stirred mixture leaves the tank at the same rate. Find an expression in terms of γ for the amount of salt in the tank any time t. also find the limiting amount of salt in the tank as t →∞ . 2.3:7) Suppose that a sum S 0 is invested at an annual rate of return r compounded continuously. (a) Find the time T required for the original sum to double in value as a function of r. (b) Determine T if r = 7% (c) Find the return rate that must be achieved if the initial investment is to double in 8 years. 2.3:16) Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton’s law of cooling. If the coffee has a temperature of 200 ° F when freshly poured, and 1 min later has cooled to 190 ° F in a room at 70 ° F, determine when the coffee reaches a temperature of 150 ° F. 2.3:18) Consider an insulated box ( a building, perhaps ) with internal temperature u(t). According to Newton’s law of cooling, u satisfies the differential equation where T(t) is the ambient (external) Temperature. Suppose that T(t) varies sinusiodally; for example, assume that T(t) = T 0 + T 1 cos ϖ t. (a) Solve Eq. (i) and express u(t) in terms of t, k, T 0 , T 1 , ϖ . Observe that part of your solution approaches zero as t becomes large; this is called the transient part. The rmainder of the solution is called the steady state; denote it by S(t).
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