At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is $70^{\circ}$F in your house. At 9:00 pm, it is $59^{\circ}$F in the house, and you notice that it is $15^{\circ}$F outside. (a) Assume that the temperature $T$ inside your home, measured in degrees Fahrenheit, is related to the temperature outside by Newton’s law of cooling. Express this relation as a differential equation. Let t denote time, measured in hours. $\displaystyle{ \frac{dT}{dt} = }$ Use k for any constant of proportionality in your equation; your equation may involve T and the values in the problem. (b) Solve the differential equation to estimate the temperature in the house when you get up at 8:00 am the next morning. Temperature = deg. F. (c) The water pipes inside the house are well insulated from the outside, so their temperature depends only on the air temperature inside. Should you worry about these pipes freezing during the night? ? yes no (d) In your answers to parts (a) and (b), what assumption did you make about the temperature outside? ? Outside temperature is constant Outside temperature is random Outside temperature varies logarithmically with time Outside temperature varies with the season Outside temperature varies exponentially with time I made no assumption about outside temperature (e) What effect does this assumption have on the amount of cooling predicted by your model? ? This model predicts too little cooling because outside temperatures are cooler in the winter This model probably predicts too little cooling at night because outside temperatures tend to fall at night This model probably predicts too little cooling during daylignt hours because outside temperatures tend to be warmer during the day The rate of cooling in this model is correct; nothing in these calculations depends on the way that outside temperature varies during a day.