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 0.5 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 ris time. The temperature of the water is Tw in units of°F and is given by the equation below where the constant a=1/12 with units Meet. The temperature of the thermometer is Tr() has units of F. The initial temperature of the package at the surface of the ocean (when it begins to sink) is 65 °F. The package (plus thermometer) behaves as a first order system with a time constant of 45 seconds (a) Derive the equation that describes the temperature reading reported by the thermometer on the sinking package versus time
(b) Plot (overlay) the temperature of the water through which the package is passing and the reading reported by the thermometer as a function of time for 0 to 240 seconds on the same graph. Reminder: use MATLAB; include script: properly annotate (labels, titles, legend, etc.)
(c) At 30, 60 and 90 seconds after the package is released. i. Using the derived equation, calculate the temperature reading reported by the thermometer. ii. Calculate the actual temperature of the water through which the package is passing at that instant.
(d) What is the maximum temperature reading reported by the thermometer? What is the time at which that occurs to the nearest 1/100th of a second? What is the actual temperature of the water at that instant? Mark the point on the plot. /Note: you do not need to solve this part analytically. You may estimate using your calculated MATLAB plot./
(e) Calculate the final temperature reading reported by the thermometer assuming a very deep ocean. Show all work. Surface of ocean Package with Thermometer T-125-92(1-0.6e-")。F y(t) dy dt