1. Applications: Decay Find a function that satisfies dP/dt = – kP.
OBJECTIVE
Find a function that satisfies dP/dt = – kP.
Convert between decay rate and half-life.
Solve applied problems involving exponential decay.

2. 3.4 Applications: Decay The equation
shows P to be decreasing as a function of time, and the solution
Shows it to be decreasing exponentially. This is exponential decay. The amount present at time t = 0 is P0.

3. 3.4 Applications: Decay

4. 3.4 Applications: Decay THEOREM 10
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3.4 Applications: Decay
THEOREM 10
The decay rate k and the half–life T are related by

5. 3.4 Applications: Decay
Example 1: Plutonium-239, a common product of a
functioning nuclear reactor, can be deadly to people
exposed to it. Its decay rate is about % per
year. What is its half-life?

6. 3.4 Applications: Decay Quick Check 1
a.) The decay rate of cesium-137 is 2.3% per year. What is its half-life?
b.) The half-life of barium-140 is 13 days. What is its decay rate?

7. 3.4 Applications: Decay
Example 2: The radioactive element carbon-14 has a
half-life of 5730 yr. The percentage of carbon-14
present in the remains of plants and animals can be
used to determine age. Archaeologists found that the
linen wrapping from one of the Dead Sea Scrolls had
lost 22.3% of its carbon-14. How old was the
linen wrapping?

8. 3.4 Applications: Decay Example 2 (continued):
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3.4 Applications: Decay
Example 2 (continued):
1st find the decay rate, k.
Then substitute the information from the problem and
k into the equation

9. 3.4 Applications: Decay
Example 2 (concluded):

10. 3.4 Applications: Decay Quick Check 2
How old is a skeleton found at an archaeological site if tests show that it has lost 60% of its carbon-14?
First find the decay rate. We know from Example 2 that the decay rate
Then substitute the information from the problem and into

11. 3.4 Applications: Decay Quick Check 2 Concluded
Thus the skeleton is approximately 7575 years old.

12. 3.4 Applications: Decay Example 3: Following the birth of their
granddaughter, two grandparents want to make an
initial investment of P0 that will grow to $10,000 by
the child’s 20th birthday. Interest is compounded
continuously at 6%. What should the initial
investment be?
We will use the equation

13. 3.4 Applications: Decay Example 3 (continued):
Thus, the grandparents must deposit $ , which
will grow to $10,000 by the child’s 20th birthday.

14. 3.4 Applications: Decay Quick Check 3
Repeat Example 3 for an interest rate of 4%
We will use the equation
Thus the grandparents must deposit $ , which will grow to $10,000 by the child’s 20th birthday.

15. 3.4 Applications: Decay THEOREM 11
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3.4 Applications: Decay
THEOREM 11
The present value P0 of an amount P due t years later,
at an interest rate k, compounded continuously, is
given by
Note:

16. Newton’s Law of Cooling
3.4 Applications: Decay
Newton’s Law of Cooling
The temperature T of a cooling object drops at a rate
that is proportional to the difference T – C, where C is
the constant temperature of the surrounding medium.
Thus,
The function that satisfies the above equation is

17. 3.4 Applications: Decay
Example 4: A body is found slumped over a desk in a study. A coroner arrives at noon, immediately takes the temperature of the body, and finds it to be She waits 1 hr, takes the temperature again, and finds it to be She also notes that the temperature of the room is 70. When was the murder committed?

18. 3.4 Applications: Decay Example 4 (continued):
We will assume that the temperature of the body was
normal (T = 98.6°) when the murder was committed
(t = 0). Thus,
This gives

19. 3.4 Applications: Decay Example 4 (continued):
To find the number of hours, N, since the murder was
committed, we must first find k. From the two
temperature readings the coroner made, we have
Then, we can solve for k.

20. 3.4 Applications: Decay
Example 4 (continued):

21. 3.4 Applications: Decay Example 4 (concluded):
Then we can substitute back into either one of our first
equations to solve for N.

22. 3.4 Applications: Decay Section Summary
The decay rate, k, and the half–life, T, are related by
The present value of an amount p due t years later, at an interest rate k, compounded continuously, is given by P0 = Pe–kt.

23. 3.4 Applications: Decay Section Summary
According to Newton’s Law of Cooling, the temperature T of a cooling object drops at a rate that is proportional to the difference T – C, where C is the surrounding room temperature.
Thus, we have
and