Newton’s law of cooling states that the temperature of a heated object decreases exponentially over time toward the temperature of the surrounding medium. In other words,

where T( t) is the temperature of a heated object at a given time t,

S is the ambient or constant temperature of the surrounding medium,

I is the initial temperature of the heated object, and

k is a negative constant (rate): T(t) = S + (I – S)e –kt

Problem One: Cooling Time of a Pizza A pizza baked at 450F is removed from the oven at 5:00 pm into a room that is a constant 70F. After 5 minutes, the pizza is at 300F.

a) At what time can you begin eating the pizza if you want its temperature to be 135F ?

b) Determine the amount of time that needs to elapse before the pizza is 160 F.

Another Application

Problem Two: An Open and Shut Case?

The great detective Sherlock Holmes and his assistant, Dr. Watson, are discussing the murder of actor Cornelius McHam. McHam was shot in the head, and his understudy, Barry Moore, was found standing over the body with the murder weapon in hand. Let’s listen in:

Watson: Open-and-shut case, Holmes. Moore is the murderer.

Holmes: Not so fast, Watson – you are forgetting Newton’s Law of Cooling!

Watson: Huh?

Holmes: Elementary, my dear Watson. Moore was found standing over McHam at 10:06 p.m., at which time the coroner recorded a body temperature of 77.9°F and noted that the room thermostat was set to 72°F. At 11:06 p.m. the coroner took another reading and recorded a body temperature of 75.6°F. Since McHam’s normal temperature was 98.6°F, and since Moore was on stage between 6:00 p.m. and 8:00 p.m., Moore is obviously innocent. Ask any precalculus student to figure it out for you.

How did Holmes know that Moore was innocent?

Investigation of Problem Two: An Open and Shut Case?

Analytically: Using Newton’s Law of Cooling, write the exponential model of this situation

Numerically: Complete this table using the exponential model of this situation and determine the hour of death

time t,hours T, body temperature

11:06 1

10:06 0

9:06 -1

8:06

7:06

6:06

Graphically: Sketch the exponential model of this

situation and use the graph to estimate the time of death

x

y

5

:06

6

:06

7

:06

8

:06

9

:06

10

:06

11

:06

12

:06

1

:06

Analytically: Use the exponential model of this

situation to find the exact time of death

S =

I =

T(t) =

Now, use I and the second temperature reading to find k

Logistic models are exponential functions that can model situations where the growth of the dependent variable is limited. The logistic growth model, where a, b, and c are constants with c and b being positive is given as

btae

ctP

1

)(

Al Gebra has just returned home to Pythagoras, Mathechusetts (population 1000) from another city which has had influenza cases. Assume the community in Pythagoras has not had influenza shots and all are susceptible and that the spread of the disease in any community is predicted to be given by the logistic curve,

where N is the number of people who have contracted influenza after t days.

a. How many people have contracted influenza after 10 days? After 20 days?

b. How many days will it take until half the community has contracted influenza?

c. Does N approach a limiting value as t increases without bound? Explain.

tetN

3.09991

1000)(

An Application Problem

Logistic model

Analytically: Use the logistic model of this situation to answer parts a & b

Numerically: Complete this table using the logistic model of this situation

t , days T, infected people

5

25

125

625

3125

Graphically: Sketch the logistic model of this situation

Verbally: Does N approach a limiting value as t

increases without bound? Explain your answer. You may refer to the table and/or the graph.

x