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Newton's Law of Cooling

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Case: At 9:00 PM a coroner arrived at a hotel room of a murder victim. The temp. of the room was 70 degrees F. It was assumed that the victim had a body temp. of 98.6 degrees F (AT THE TIME OF DEATH)(not at 9:00PM). The coroner took the victim's temp. at 9:15 PM at which it was 83.6 degrees F and again at 10:00 PM at which it was 80.3 degrees F.
A. At what time did the victim die? I solved that question to be at 7:15 PM (165 min before 10:00PM)
B. If the assumption of the victim's body temp. at the time of death was found to be incorrect due to a major outbreak of influenza at the hotel during the victim's stay and the usual fever a flu case will run is 103 degrees F. What is the time of death then? I solved it to be 6:37 PM(203 min before 10:00PM).
C. If the victim had no influenza at the time of death, but the room temp was really 75 degrees F what was the time of death then? answer was 7:47PM(133 min before 10:00PM)
D. Assuming the victim was murdered 3 minutes of the time the murderer got off the elevator(which had video surveillance) what five minute sequence should have been reviewed in each scenarios a, b, and, c?

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Theory
Newton's Law of Cooling describes the cooling of a warmer object to the cooler temperature of the environment. Specifically we write this law as,
T (t) = Te + (T0 − Te ) e - kt (1)
where T (t) 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.
Solution
1. In this case: Te ...

Solution Summary

The solution provides detailed analysis, theory and discussion for the Newton's Law of Cooling word problem.

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Newton's Law of Cooling explained

Background:

In the beginning of the year, we discussed qualitative graphs and independent/dependent variables. One of the situations we graphed was the temperature of some food taken out of the oven while we waited for it to cool enough to put into the refrigerator. This situation can actually be modeled with an exponential equation of the form y = ab (the standard form of an expontial equation).
The equation we use is called ''Newton's law of cooling. This law models the temperature of an object as it cools down, as when a pizza is removed from the oven and placed on the kitchen counter. The function model is T(x) = T + (T - T ) e , k<0. In this equation,

T = the room temperature, or the temperature of the surrounding medium (degrees F)

T = the initial temperature of object (oven temperature when pizza was baked) ( F)

K = is the cooling rate constant as determined by the nature and physical properties of the object (in this case, the pizza). Note: k is always less than zero.

X = the time (in minutes) after the pizza is removed from the oven and placed at room temperature

T(x) = the temperature of the pizza at x minutes after the pizza is taken out of the oven.

Problem:

John and Sally are making pizzas for a late night snack while studying for their Math 107 final. John has a meat lovers pizza, and Sally has double cheese. The two pizzas are taken from a 450 degree oven and placed on the counter to cool. The temperature in the kitchen is 75 degree, and the cooling rates for the pizza are:

Meat Pizza: k= -.29 Cheese Pizza: k = -.35

1. Sally waits ten minutes after the pizza has come out of the oven, and then takes a big bite from her pizza. Did she wait long enough? (Most people like to eat their pizza when it is 110 F.)

2. How long should John wait before digging into his?

3. Make a graph showing the cooling rates for each pizza for 30 minutes , and illustrate on the graph when a pizza is:

a) Just cool enough not to burn your mouth (about 130 degrees)
b) When it is too cool to be enjoyed (less than about 90 degrees)
c) And state on the graph the range of time someone should wait to eat the pizza after it comes out of the oven.

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