本题目来源于试卷: Differential Equations，类别为 IB数学
[问答题]
A tank has been prepared h a;k7 j y4c,.tue0sj9x zadct;sw3g6 in06 0ehzg cdp6ley9)*kz(wrc0h6r ovq order to mix a color for a fabric dyeing process. The tank initially contains water. A color concentrate is a premix of color powder and a small amount of water The color concentrate begins to flow into the tank. The color solution is kept uniform by stirring and leaves the tank through ah)0dqz p6c(6kry6e9rhe0o *wzvg cl0n outlet at its base. Let x grams represent the amount of the color powder in the tank and let t minutes represent the time since the color concentrate began flowing into the tank.
The rate of change of the amount of color powder in the tank, $\frac{\mathrm{d} x}{\mathrm{~d} t}$ , is described by the differential equation
$\frac{\mathrm{d} x}{\mathrm{~d} t}=4 e^{-\frac{t}{5}}-\frac{x}{t+3}$
1. Show that t+3 is an integrating factor for this differential equation.
2. Hence, by solving this differential equation, show that $ x(t)=\frac{160-20 e^{-\frac{t}{5}}(t+8)}{t+3}$ .
3. Sketch the graph of x versus t for $0 \leq t \leq 50 $ and hence find the maximum amount of color powder in the tank and the value of t at which this occurs.
4. Find the value of t at which the amount of color powder in the tank is decreasing most rapidly.
The rate of change of the amount of color powder leaving the tank is equal to
5. Find the amount of color powder that left the tank during the first 50 minutes.
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