A package with a thermometer attached to the outside is released at the surface of the ocean (see sketch below). The package sinks with a constant velocity of 0.3 fi/sec. The package starts sinking when released at the surface of the ocean. The displacement of the package is defined by y(t) which has units of feet, and t is time. The temperature of the water is T in units of °F and is given by the equation below where the constant a-1/8 with units Ifeet. The temperature of the thermometer is Tt(t) has units of"F. The initial temperature of the package at the surface of the ocean (when it begins to sink) is 68 °F. The package (plus thermometer) behaves as a first order system with a time constant of 30 seconds. Required:
Derive the equation that describes the temperature reading reported by the thermometer on the sinking package versus time.