The differential equation below models the temperature of a 94°C cup of coffee in a 24°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 74°C. Solve the differential equation to find an expression for the temperature of the coffee at time t. (Let y be the temperature of the cup of coffee in °C, and let t be the time in minutes, with t = 0 corresponding to the time when the temperature was 94°C.) 1 dy (y- 24) dt 50 Step 1 dy We have dt (y-24), 50 Which separates to become dy = -dt Step 2 Ignoring the constant of integration, integrating the left side gives (remember to use In Ju| where appropriat S dy = y-24 Step 3 Ignoring the constant of integration, integrating the right side gives Sat = dt3D 50 Step 4 We now have Inly - 24| = -t + C.and so ly – 24| = 50 Step 5 Consequently, we write y-K +24, where K- te Step 6 Finally, using y (0) -94 allows us to solve for K- Step 7 The temperature of the coffee at the time is described by the following equation.

A dead body was discovered shortly before midnight. At exactly midnight, the temperature of the body was 85°F. Four hours later, the body had cooled to 74°F. If the room where the body was found was a constant 68°F, what was the time of death? Solve this problem by first setting up the differential equation that describes this situation, then find its solution to answer the question. Note: Newton's Law of coaling states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings. (Take normal body temperature to be 98.6°F)

All temperatures are in degrees Fahrenheit. A cup of coffee at 173 degrees is poured into a mug and left in a room that stays at 68 degrees. After 2 minutes, the coffee is 132 degrees. Assume that the differential equation describing Newton's Law of Cooling is (in this case) dT = k(T - 68) dt What is the temperature of the coffee after 12 minutes? After how many minutes will the coffee be 100 degrees?

Solve the differential equation to find velocity v as a function of time t if v = 0 when t = 0. The differential equation models the motion of two people on a toboggan after consideration of the forces of gravity, friction, and air resistance. dv 12.5 = 43.6 – 1.25v dt