A curve passes through the point (0,5) and has the property that the slope of the curve at every point P is four times the y-coordinate of P. What is the equation of the curve? y(x) = ngs 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. lems Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. Let k > 0 be the constant of proportionality. Assume the coffee has a temperature of 205 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 189 degrees in a room at 64 degrees. (a) Write an initial value problem for the temperature T of the coffee, in Fahrenheit, at time t in minutes. Your answer will contain the uknown constant k: T(0) (b) Solve the initial value problem in part (a). Your answer will contain the unknown constant k. T(t)= (c) Determine the value of the constant k 11 n2... dT dt 3 m4 m5 m6 m 7 k m 8 m 9 em 10 em 11 ettings oblems em 1 em 2... em 3 em 4 lem 5 lem 6 lem 7 lem 8 lem 9 lem 10 (d) Determine when the coffee reaches a temperature of 159 degrees. minutes. A tank contains 60 kg of salt and 1000 L of water. Pure water enters a tank at the rate 8 L/min. The solution is mixed and drains from the tank at the rate 4 L/min. (a) Write an initial value problem for the amount of salt, y, in kilograms, at time t in minutes: dy =(kg/min) y(0) dt kg. (b) Solve the initial value problem in part (a). y(t)= kg. (c) Find the amount of salt in the tank after 3.5 hours. amount= (kg) (d) Find the concentration of salt in the solution in the tank as time approaches infinity. (Assume your tank is large enough to hold all the solution.) concentration = (kg/L)