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Part I: A Cup of Hot CoffeeSir Isaac Newton found that the temperature of something heated will cool down at different rates, depending on the temperature of the environment in which it is cooling. The “Newton’s Law of Cooling” equation was derived based on this function:Tt=Te+(T0-Te)e-ktwhere Tt is the temperature of the object at time t, Te is the constant temperature of the environment, T0 is the initial temperature of the object , and k is a constant that depends on the material properties of the object.Look at the statement about k. It is saying that k is a constant that depends on the material. Can you think of two liquids that would cool at different rates? What physical properties of the two liquids make that happen? a) In a 72℉ room, your 180℉ coffee will be 150℉ after two minutes. Using this information and the functionabove, write an equation that can be used to find k. Use this equation to show algebraically that k≈0.1627112002. Clearly show all steps! (Hint: use the “STO→” key to save k in your calculator—you’ll need it in the next part!) You’ve discovered that the perfect temperature for coffee is 120℉. Write an equation using the provided information that can be solved to find how long you need to wait for the perfect cup of coffee! Use this equation to show algebraically that t≈4.98 minutes. Clearly show all steps! -9525019177000How long will it take for your cup of coffee to cool down to 75℉? What is the temperature of the coffee after 30 minutes? Part II: A Dead Body! (Well that escalated quickly…)Detective Barbara Boddy is called to the scene of a crime in a college science lab where the dead body of an unnamed chemistry student has just been found in a closet. It is clear the body was there for some time—possibly even while students were working in the lab the previous night. The detective arrives on the scene at 5:41am and begins her investigation. Immediately, the temperature of the body is taken and is found to be 78.0℉. The detective checks the programmable thermostat and finds that the lab has been kept at a constant 71℉ for several days.After evidence from the crime scene is collected, the temperature of the body is taken once more and found to be 76.6℉. This last temperature reading was taken exactly one hour after the initial reading.588645019685000Based on the key-card entry records, it is clear that there were only four students in the lab the night before:The dead chemistry student arrived at the lab at 7pm the previous night and never left the lab.Vivienne Scarlet got into the science lab at 6pm and left at 10pmMichael Mustard got into the science lab at midnight and worked until 2am.Elizabeth Peacock got into the science lab at 10pm and worked until midnight.The next day Detective Boddy is asked by another investigator, “What time did our victim die?” Assuming that the victim’s body temperature was normal (98.6℉) prior to death, what is her answer to this question? Newton’s Law is how detectives determine time of death! Solve the crime. Find the dead student’s constant of cooling k, her time of death, and the name of the murderer! ................
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