A friend of mine lives on the coast near the ocean and frequently likes to eat lunch on the beach. Out above the ocean, the temperature remains nearly constant throughout the day at about 20°C. The temperature at their house 50 km away from that point, however, varies throughout the day. In the morning, the temperature varies linearly for the first 5 hours of the day, increasing from 20°C to 30°C.
(a) If the temperature also varies linearly from their house to the point above the ocean that is at constant temperature, derive an equation to describe the temperature field as a function of position and time. Assume the two points are at the same elevation.
(b) Based on this function, find the equation to determine the total rate of change of temperature as a function of position and time.