Suppose you have just poured a cup of freshly brewed coffee with temperature 95∘c in a room where the temperature is 20∘c. newton's law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. therefore, the temperature of the coffee, t(t), satisfies the differential equationdtdt=k(t−troom)where troom=20 is the room temperature, and k is some constant. suppose it is known that the coffee cools at a rate of 1∘c per minute when its temperature is 70∘c. a. what is the limiting value of the temperature of the coffee? limt→[infinity]t(t)=equation editorequation editorb. what is the limiting value of the rate of cooling? limt→[infinity]dtdt=equation editorequation editorc. find the constant k in the differential equation. k=equation editorequation editor. d. use euler's method with step size h=3 minutes to estimate the temperature of the coffee after 15 minutes. t(15)=equation editorequation editor.

Aheat engine running backward is called a refrigerator if its purpose is to extract heat from a cold reservoir. the same engine running backward is called a heat pump if its purpose is to exhaust warm air into the hot reservoir. heat pumps are widely used for home heating. you can think of a heat pump as a refrigerator that is cooling the already cold outdoors and, with its exhaust heat qh, warming the indoors. perhaps this seems a little silly, but consider the following. electricity can be directly used to heat a home by passing an electric current through a heating coil. this is a direct, 100% conversion of work to heat. that is, 19.0 \rm kw of electric power (generated by doing work at the rate 19.0 kj/s at the power plant) produces heat energy inside the home at a rate of 19.0 kj/s. suppose that the neighbor's home has a heat pump with a coefficient of performance of 4.00, a realistic value. note: with a refrigerator, "what you get" is heat removed. but with a heat pump, "what you get" is heat delivered. so the coefficient of performance of a heat pump is k=qh/win. an average price for electricity is about 40 mj per dollar. a furnace or heat pump will run typically 200 hours per month during the winter. what does one month's heating cost in the home with a 16.0 kw electric heater? what does one month's heating cost in the home of a neighbor who uses a heat pump to provide the same amount of heating?