According to the weather forecast, tomorrow the temperature in Sudbury will incr

Question

According to the weather forecast, tomorrow the temperature in Sudbury will increase from -10 degree C at 8:00 AM to 0 degree C at 16:00 PM, and the temperature in Toronto will change from -5 degree C at 8:00 AM to 5 degree C at 16:00 PM. Consider the distance between Sudbury and Toronto as 400 km, we can approximate the temperature variation T(x,t) by T(x,t) = 0.0125.v +1.25t - 20, where x represents the location (x = 0 for Sudbury and .x = 400 for Toronto), and t is time (t = 8 at 8:00 AM and t = 16 at 16:00 PM). Assume you are leaving Sudbury at 10:00 AM to Toronto and driving at a speed of 100 km/hr, please find the temperature increase rate you will experience on your trip using the material derivative (15 marks) and the local temperature difference at the beginning and end of your trip and the time you spend on the trip (10 marks).

Explanation / Answer

speed = 100km/h so dx/dt = 100 km/h a) dT/dt = rate of increase of temperature with time = 0.0125 dx/dt + 1.25 dT/dt = 0.0125*100 + 1.25 = 2.50 degrees/hour b) local temperature at beginning = 0 + 1.25*10 - 20 = --7.5 degrees time taken to reach toronto = 400/100 = 4 hours at end = 0.0125*400 + 1.25*14 - 20 = 5 + 17.5 - 20 = 2.5 degrees Therefore difference = 10 degrees C

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