MATH241 Lecture 20: Exponential Growth and Decay (continued)
9 May 2018
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Professor
MATH241 - Lecture 20 - Exponential Growth and Decay (continued)
3.8: Exponential Growth and Decay (continued)
II. Radioactive Decay
Let initial mass of a radioactive isotopem0=
Then mass remaining at time m(t) = t
The relative decay rate is constant, we can model radioactive decay using the K=1
m dt
dm
differential equation m
dt
dm =K
Radioactive Decay Model
em (t) = m0Kt
with relative decay rate K< 0
Notice that lim
t → ∞ m(t) = m0lim
t → ∞ eKt = 0
Half Life
The time required for half of any given quantity to decay m T
(2
1)=2
m0
Relative Decay Rate and Half Life
e m T
(2
1)=m0
KT
2
1=2
m0
→ eKT
2
1=2
1
→ T n K
2
1=l2
1
→ ln K=1
T
2
1(2
1)
OR ln 2K= − 1
T
2
1
The units are 1
years
Notice that since K< 0 T
2
1> 0
Example
:
The half life of Nickel-63 is 100 years
A) Find the decay rate
B) Find a formula for the mass of the sample if the initial mass is 320 mg
C) Find the mass remaining after 500 years
D) When will the mass be reduced to 30 mg?
Solution:
A) em (t) = m0Kt
em (100) = m0100K=2
m0
→ e100K=2
1
→ 00K n 1 = l(2
1)
→ ln 2K= − 1
100
B) 20em (t) = 3 − ln 2
t
100
C) 20em (500) = 3 − ln 2
100
500
20e= 3 −5 ln 2
20e= 3 ln 2−5
20 = 3 2
(−5)
20 = 3 (1
32 )
mg0= 1
D) 20e0m(t) = 3 − ln 2
t
100 = 3
→ e− ln 2
t
100 =30
320 =3
32
→ ln 2 n − t
100 =l3
32
→ ln t= − ln 2
100 (3
32 )
ln = ln 2
100 (3
32 )
Newton’s Law of Cooling
The rate of cooling of an object is proportional to the temperature difference between the
object and its surroundings
Document Summary
Math241 - lecture 20 - exponential growth and decay (continued) Let initial mass of a radioactive isotope mass remaining at time t. The relative decay rate is constant, we can model radioactive decay using the dm. K = 1 m dt dm = k m dt differential equation. Kt m (t) = m0 with relative decay rate k < 0. Notice that lim t m (t) = m0 lim t ekt = 0. The time required for half of any given quantity to decay m t( 2: = 2 m0. 20e m (t) = 3 ln 2 ln 2 t. The rate of cooling of an object is proportional to the temperature difference between the object and its surroundings temperature of the object at time t. T 0 = temperature of the surroundings initial temperature of the object dt dt is proportional to the temperature difference (t. The rate of cooling dt = k (t dt.
