1. 𝑷= 𝑷 𝟎 ⅇ 𝒓𝒕 𝑷 𝒕 = 𝑷 𝟎 𝟏+ 𝒓 𝒏 𝒏𝒕 EXPONENTIAL AND LOGISTIC MODELING
1)Interest
𝑷 𝒕 = 𝑷 𝟎 𝟏+ 𝒓 𝒏 𝒏𝒕
Compounded n times per year for t years
𝑷= 𝑷 𝟎 ⅇ 𝒓𝒕
Continuously Compounded

2. EX#1: How long does it take to double $1000 at an annual interest rate of 6.35% compounded monthly?
𝑷 𝒕 = 𝑷 𝟎 𝟏+ 𝒓 𝒏 𝒏𝒕
Given :
Po=$1000
R= 6.35% =
N=12
P(t)= 2000
2000=𝟏𝟎𝟎𝟎 𝟏+ .𝟎𝟔𝟑𝟓 𝟏𝟐 𝟏𝟐𝒕
2= 𝟏+ .𝟎𝟔𝟑𝟓 𝟏𝟐 𝟏𝟐𝒕
log 2= 𝒍𝒐𝒈 𝟏+ .𝟎𝟔𝟑𝟓 𝟏𝟐 𝟏𝟐𝒕
log 2 log =12𝑡
log 2 =12𝑡⋅ log
𝑡≈10.9 𝑦𝑒𝑎𝑟𝑠
log log =𝑡

3. Exponential Growth/Decay Problems
𝑃= 𝑃 𝑜 𝑒 𝑘𝑡
Here 𝑃 𝑜 is the initial amount and k is the exponential growth/decay rate.
If k is positive then we will have a growth model and if k is negative then we will have a decay model.

4. EX #2: Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 250,000? Given k=
𝑃= 𝑃 𝑜 𝑒 𝑘𝑡
Know:
250,000=100 𝒆 𝟎.𝟔𝟗𝟑𝟏𝑡
𝑃 𝑜 =100
𝑃=250,000
𝑙𝑛 2500 = 𝑙𝑛 𝑒 𝑡
𝑘=0.6931
𝑙𝑛 2500 = 𝑡 l𝑛 𝑒
l𝑛 =𝑡
𝑡=11.29 ℎ𝑜𝑢𝑟𝑠

5. EX. 3 Suppose the half-life of a certain radioactive substance is 40 days and there are 8 grams present initially. Find the time when there will be 1 g (gram) of the substance remaining. K=
𝑃= 𝑃 𝑜 𝑒 𝑘𝑡
1=8 ⅇ −0.6931𝑡
ln − =𝑡
𝑡=120 𝑑𝑎𝑦𝑠
1 8 = ⅇ −0.6931𝑡
ln 1 8 = 𝑙𝑛ⅇ −0.6931𝑡
𝑡=3 ℎ𝑎𝑙𝑓−𝑙𝑖𝑓𝑒𝑠

6. 𝑨= 𝑨 𝟎 ⅇ 𝒌𝒕 Uninhibited Growth ln 8 5 =𝑘 ln ⅇ 800=500 ⅇ 𝑘(1) 𝑘=0.47000
𝑨= 𝑨 𝟎 ⅇ 𝒌𝒕
EX. 4 A culture of bacteria follows the law of uninhibited growth. If 500 bacteria are present initially and there are 800 after 1 hour. (Hint: Find k to 5 decimal places)
A. How many will be present in the culture after 5 hours?
B. How long before the culture contains 20,000 bacteria?
𝐴= 𝐴 0 ⅇ 𝑘𝑡
ln =𝑘 ln ⅇ
800=500 ⅇ 𝑘(1) 𝑘=
8 5 = ⅇ 𝑘
ln =𝑘

7. A culture of bacteria follows the law of uninhibited growth
A culture of bacteria follows the law of uninhibited growth. If 500 bacteria are present initially and there are 800 after 1 hour.
A. How many will be present in the culture after 5 hours?
A=500 ⅇ (0.4700)(5)
A=500 ⅇ (2.35)
𝐴=5,
B. How long before the culture contains 20,000 bacteria?
20,000=500 ⅇ (0.4700)(𝑡)
= ⅇ (𝑡)
40= ⅇ (𝑡)
ln =𝑡
ln 40 =0.4700𝑡 ln ⅇ
𝑡=7.8487

8. Your Turn
The streptococci bacteria population N at time t (in months) is given by N = N0e2t where N0 is the initial population. If the initial population was 100, how long does it take for the population to reach one million?
N = N0e2t
ln10,000 = ln e2t
1,000,000 = 100e2t
𝑡=4.6054
ln10,000 = 2t ln e

9. Richter Scale Model – measures magnitude R of an
earthquake, a is amplitude in micrometers of the vertical
ground motion at the receiving station, T is period of the
seismic wave in seconds, and B is the dampening of the
seismic wave as the distance from the epicenter
increases

10. Richter Scale Intensity
How many times more severe was the 2001 earthquake in Gujarat, India (R1 = 7.9) than the 1999 earthquake in Athens, Greece (R2= 5.9) ?
𝑅= 𝐼 𝐼 𝑜

11. pH Scale Model – a scale that measures how acidic or basic a substance ranks. The scale ranges from 0 to 14, where 7 represents neutral and each whole pH value below 7 is ten times more acidic than the next higher value

12. Newton’s Law of Cooling – The temperature T of an object at time t, where Tm = temperature of the surrounding medium, and T0 = initial temperature of the object.

13. A hard-boil egg at a temperature of 96 ◦ C is placed in 16 ◦ C water to cool . Four minutes later the temperature of the egg is 45 C. Determine when the egg will be 20 ◦ C
45=16+(96−16) 𝑒 −4𝑘
29 80 = 𝑒 −4𝑘
𝑙𝑛 =−4𝑘
𝑇 𝑜 =96 𝐶
𝑇 𝑚 =16 C
𝑘=0.254
𝑇(4)=45 C
𝑙𝑛 =𝑙𝑛 𝑒 −4𝑘
20=16+(96−16) 𝑒 −0.254𝑡
11.8

14. Volume of Sound Model
the volume L is measured in decibels (db) and I is the intensity in watts per square meter (W/m2 ).

15. Logistic Growth Function
where the constant c is the limit to growth.

16. 𝑆 𝑡 = 1500 1+ 38𝑒 −0.9𝑡 t = 0 is the day the rumor begins
Surfside High School has 1500 students. Max, Patti, Juan, and Andrea start a rumor, which spreads logistically so that
models the number of students who have heard the rumor by the end of t days, where
t = 0 is the day the rumor begins
𝑆 𝑡 = 𝑒 −0.9𝑡
A. How many students have heard the rumor by the end of day 0? 38
B. How long does it take for 1000 students to hear the rumor?
4.8 days

17. The function: 𝐹 𝑡 = 64, 𝑒 −1.2𝑡
describes the number of people F t who have become ill
with a flu breakout 𝑡 weeks after its initial discovery within a town of 64,000 people
A. What is the limiting size of , the population that can become ill?
B. How many people were ill at the beginning of the epidemic?
C. How many people were ill by the end of the
fourth week?