1. 3.5 Exponential Growth and Decay
1.) In 1970, the U.S. population was million. By 2007, it had grown to million.
a.) Use the exponential growth model to find an exponential growth function that models the data for 1970 through 2007.
A = A0ekt
Plug in to find k:
300.9 = 𝑒 37𝑘
2) Solve:
= 𝑒 37𝑘
3) k = .0106
4) Growth Model:
A = 𝑒 .0106𝑡
ln = 37𝑘 𝑙𝑛𝑒
b.) By which year will the U.S. population reach 315 million?
2) Solve: = 𝑒 .0106𝑡
ln = .0106𝑡
3) t = 41.3 years
1) 315 = 𝑒 .0106𝑡
≈ 2011

2. 2.) In 1990, the population of Africa was 643 million and by 2006 it had grown to 906 million.
a.) Use the exponential growth model A = A0ekt, in which t is the number of years after 1990, to find the exponential growth function that models the data.
b.) By which year will Africa’s population reach 2000 million, or 2 billion?
Plug in to find k:
906 = 643 𝑒 16𝑘
2) k = .0214
3) Growth Model:
A = 643 𝑒 .0214𝑡
2) Solve: = 𝑒 .0214𝑡
1) 2000 = 643 𝑒 .0214𝑡
3) t = 53 years
4) ≈ 2043
ln = .0214𝑡

3. 3a.) Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14.
What do we know?
Therefore, there will be half of the original amount left after 5715 years
.5A0 = A0e5715k
The half life must be 5715 years
.5 = e5715k
ln .5 = 5715k lne
A =A0e t
k =
b.) In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls.
.76A0 = A0e t
.76 = e t
ln .76 = t
t = 2268 years old
That was in 1947 so we must add 68 years to get to Total age is 2336 years.

4. 4. ) Strontium-90 is a waste product from nuclear reactors
4.) Strontium-90 is a waste product from nuclear reactors. As a consequence of fallout from atmospheric nuclear tests, we all have a measurable amount of Strontium-90 in our bones.
a.) The half-life of strontium-90 is 28 years, meaning that after 28 years a given amount of the substance will have decayed to half the original amount. Find the exponential decay model for strontium-90.
b.) Suppose that a nuclear accident occurs and releases 60 grams of strontium-90 into the atmosphere. How long will it take for strontium-90 to decay to a level of 10 grams.
.5A0 = A0e28k
A =A0e-.0248t
.5 = e28k
k =
10 = 60e-.0248t
ln = t lne
t = 72.2 years old

5. Logistic Growth Models
Using the logistic growth model from your notes, explain what happens as t →∞ and how this leads to a horizontal asymptote. What is the horizontal asymptote?
𝐴= 𝐶 1+𝑎 𝑒 −𝑏𝑡
Hint: What does 𝑎 𝑒 −𝑏𝑡 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ 𝑎𝑠 𝑡→∞
As 𝑡→∞, 𝑎 𝑒 −𝑏𝑡 →0
Therefore, as 𝑡→∞, the logistic growth model approaches:
A = 𝐶 1+0
A = 𝐶 1
This means that y = c must be a horizontal asymptote.
Therefore, A can never exceed C.
C is called the limiting size of A
A = C

6. The function
f(t) = ,000
1 + 20e-1.5t
describes the number of people, f(t), who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants.
How many people became ill with the flu when the epidemic began?
f(0) ≈ 1428 people
b.) How many people were ill by the end of the fourth week?
f(4) ≈ 28,582 people
c.) What is the limiting size of f(t), of the population that becomes ill?
30,000 people

7. 2.) In a learning theory project, psychologists discovered that
f(t) = ____0.8_____
1 + e-0.2t
is a model for describing the proportion of correct responses, f(t), after t learning trials.
Find the proportion of correct responses prior to learning trials taking place.
b) Find the proportion of correct responses after 10 learning trials.
c) What is the limiting size of f(t), the proportion of correct responses, as continued learning trials take place?
f(0) = 0.4 or 40%
f(10) = or 70%
0.8 or 80%

8. 𝑙𝑛 1 7 = -.0231t 1 7 = 𝑒 −.0231𝑡 Newton’s Law of Cooling
A cake removed from the oven has a temperature of 210oF. It is left to cool in a room that has a temperature of 70oF. After 30 minutes, the temperature of the cake is 140oF.
Use Newton’s Law of Cooling to find a model for the temperature of the cake, T, after t minutes.
70 = 140 𝑒 30𝑘
ln 0.5 = 30k lne
T = C + (T0 – C) 𝑒 𝑘𝑡
T = 𝑒 −.0231𝑡
0.5 = 𝑒 30𝑘
k =
140 = 70 + (210 – 70) 𝑒 30𝑘
b) What is the temperature of the cake after 40 minutes?
c) When will the temperature of the cake be 90oF?
T = 125.6°
𝑙𝑛 1 7 = t
90 = 𝑒 −.0231𝑡
20 = 140 𝑒 −.0231𝑡
1 7 = 𝑒 −.0231𝑡
t = 84 minutes

9. 2. ) An object is heated to 100oC
2.) An object is heated to 100oC. It is left to cool in a room that has a temperature of 30oC. After 5 minutes, the temperature of the object is 80oC.
Use Newton’s Law of Cooling to find a model for the temperature of the object, T, after t minutes.
What is the temperature of the object after 20 minutes?
When will the temperature of the object be 35oC?
ln 5 7 = 5k
T = C + (T0 – C) 𝑒 𝑘𝑡
50 = 70 𝑒 5𝑘
T = 𝑒 −.0673𝑡
5 7 = 𝑒 5𝑘
k =
80 = 30 + (100 – 30) 𝑒 5𝑘
T = 48.2°
t = 39 minutes