According to Newton's Law of Cooling, the rate of change of the temperature of an object can be modeled using the following differential equation: dT — к(Т — А); dt where T is the temperature of the object (in °F), A is the room temperature (in °F), and k is the constant of proportionality. (a) Use an appropriate method to solve the differential equation. Remember that k and A are constants. (b) Use your answer to (a) to solve the following: On a crime show, a dead body is discovered in a hotel room. A forensic technician recorded that the victim's body temperature was 91.4°F at 6:00 pm. One hour later, the coroner arrived and found that the victim's body temperature had fallen to 88.7°F. If the thermostat in the hotel room is set at 68° F, determine when the victim was murdered. (Assume the victim was a healthy 98.6° F at the time of death.)

Determine the value of x for which ƒ(x) = x³ – 9x² + 5x − e7 + 11″ has the lowest instantaneous rate of change; also, give the instantaneous rate of change at that point. x = rate =

A model for the position x(t) of an Olympic jump skier says that the acceleration a(t) = x" is proportional to the square of the velocity v(t) = x'(t). Note this model comes from taking into account the friction of air against the skier, and we are greatly simplifying the problem by not taking into account gravity. In this problem, units of distance are in meters and units of 1. time are in seconds. (a) Give an ODE in terms of v(t) only (meaning: no x or a) that corresponds to this model, and find its general solution v. (b) Recalling that our model comes from taking friction into account, which will slow down the skier, say whether you expect your proportionality constant from la to be positive or negative. (c) Use your solution v(t) from la to find x(t). (d) Use the initial conditions' x(0) = 0 and v(0) = 100 to find the unique solution for x(t), the position of the skier. (e) Assume for simplicity the following information: the skiers take 8 seconds before reaching the ground after…

b) The general form of Newton's Law of cooling is: T(t) = Ta + (T(0) - Ta)e-kt where I is the temperature at any time, t. Ta is the surrounding ambient temperature and k is the cooling constant. Consider a cup of coffee at an initial temperature, T(0) of 80°C placed into the open air at 15°C. After 5 minutes the coffee cools to 65°C. Using these initial conditions: i) Calculate the cooling constant, k ii) How long will it take for the coffee to reach 20°C?