![Page 1: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/1.jpg)
Rates of Growth & Decay
![Page 2: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/2.jpg)
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![Page 3: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/3.jpg)
Example (1)
The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate of increase of the colony at any moment is directly proportional to its size ( the rate of the growth of the bacteria population is constant), find:1. The size of the colony at 3pm.2. The time it takes the colony to quadruple in size.3. Find the (absolute) growth rate function4. How fast the size of the colony was growing at 12 noon.
![Page 4: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/4.jpg)
Solution
8656
3
156
36
3)2ln(33
1
3
1
6
63
)3(66
6
6
6
)10(32)10(3200)2()10(100
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)10(100
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![Page 5: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/5.jpg)
hourbacteriaydt
dy
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4
![Page 6: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/6.jpg)
Example (2) - a
A mass of a radioactive element has decreased from 200 to 100 grams in 3 years. Assuming that the rate of decay at any moment is directly proportional to the mass( the relative rate of the decay of the element is constant), find:1. The mass remaining after another15 years.2. The time it takes the element to decay to a quarter of its original mass.3. The half-life of the element3. Find the (absolute) growth decay function4. How fast the mass was decaying at the twelfth year.
![Page 7: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/7.jpg)
Solution
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yearsinttimeamassthebetyLet
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)2
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2
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3
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200
100
200)3(100
&200)(:
100)3(
200)0(
)0()(
)()(
3
15
3
3)
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1ln(
33
1
3
13
)3(
6
3
![Page 8: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/8.jpg)
yeargydt
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![Page 9: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/9.jpg)
Example (2) – b
A mass of a radioactive element has decreased from 1000 g to 999 grams in 8 years and 4 months. Assuming that the rate of decay at any moment is directly proportional to the mass( the relative rate of the decay of the element is constant), find the half-life of he element.
![Page 10: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/10.jpg)
Solution
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tkt
t
k
k
kt
ty
ee
ktk
ke
eysoand
yearyearmonthsandyearthatnote
yandyhaveWe
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yearsinttimeatmassthebetyLet
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100
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25
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![Page 11: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/11.jpg)
yearsT
T
yTyy
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sizeitsofhalftodecayoelementtheofmassanytakesittimetheis
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0
![Page 12: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/12.jpg)
Note
We can find the half-life T1/2 in terms of the constant k or the latter in terms of the former ( T1/2 = ln2/k Or k = ln2 / T1/2) and use that to find k when T1/2 is known or find T1/2 when k is known.
![Page 13: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/13.jpg)
57733.5773)10(2006.1
2ln
1.2006(10)
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1000ln
5
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5
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:
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khadWe
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problemlasttheinThuskk
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kTeeyTyy
Then
sizeoriginalitsofhalftodecaytomasstheforneededtimethebeTLet
lifehalftheandknttaconsthebeweenprelaionshitheDeducing
kTkT
Deducing the Relationship Between half-life T1/2 and the constant k
![Page 14: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/14.jpg)
Carbon Dating
• Carbon (radiocarbon) dating is a radiometric dating technique that uses the decay of carbon-14 (C-14 or 14C) to estimate the age of organic materials or fossilized organic materials, such as bones or wood.
• The decay of C-14 follows an exponential (decay) model.
• The time an amount of C-14 takes to decay by half is called the half-life of C-14 and it is equal about 5730 years. Measuring the the remaining proportion of C-14, in a fossilized bone, for example to the amount known to be in a live bone gives an estimation of its age.
![Page 15: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/15.jpg)
Example (3)
It was determine that a discovered fossilized bone has 25% of the C-14 of a live bone. Knowing that the half-life of C-14 is approximately 5730 years and that its decay follows exponential model, estimate the age of the bone.
Solution:
)5730(000
2
1
0
0
2
1
2
1
,573014
)(,
14
kkT
kt
eyeyy
thenTCoflifehalftheSince
eytyThen
yboneliveainCofamounttheLet
![Page 16: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/16.jpg)
yearst
ytyy
thenboneliveainyamounttheofhaditfoundwasbonethettimetheatSince
yeyeyty
tkt
kke
eyeyy
t
t
t
t
kt
t
k
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t
11460)5730(25730
22
1
2
1
2
1
4
1
2
1)(
100
25
.,%25
2
1)(
2
1ln
2
1ln
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573021
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2
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1
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5730
00
0
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00
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1
5730
1
5730
)5730(000
5730
2
1
![Page 17: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/17.jpg)
Pharmacokinetics
• Pharmacokinetics (PK) is a branch of pharmacology concerned with knowing what happens to substances ( such as drugs, food or toxins) administered to a living body.
• This includes understanding the process by which such substance is assimilated, eliminated or affected by the body.
• With some exceptions ( such as in the case of liquor) the absorption of drugs follows an exponential (decay) model.
![Page 18: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/18.jpg)
Example (4)
Two doses of 32 mg of a drug with a half-life of 16 hours were administered to a patient. The second was administered 64 hours after the first.a. How long would it take the drug to reach 12.5% of its first dose.B. How long would it take the drug after the second dose to reach 8.5 mg
![Page 19: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/19.jpg)
Solution
16
2ln2ln:
2
1)(
2
1ln
2
1ln
2
1ln
1621
ln16
2
1ln
2
1
2
1
,16
32)(:
32
)(
)()(
2
1
16
02
1ln
00
1616
1
16
1
)16(
)16(000
2
1
0
0
16
2
1
TkformulathengusibykfoundcouldWeNote
yeyeyty
tkt
kke
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yand
eyty
hoursinttimeabodytheindrugthebetyLet
t
kt
t
k
kkT
kt
kt
t
![Page 20: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/20.jpg)
hoursTyTyy
amountoriginalitsofreachtodosefirstthetakesittimethebeTLet
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02
1ln
00
16
![Page 21: Rates of Growth & Decay. Example (1) The size of a colony of bacteria was 100 million at 12 am and 200 million at 3am. Assuming that the relative rate](https://reader036.vdocuments.site/reader036/viewer/2022062802/56649eb35503460f94bba937/html5/thumbnails/21.jpg)
dosefirstthetakingfromhoursaftersThat
dosecondsethetakingfromhoursaftermgreachesdrugtheofamounttheThus
tt
tZ
thenmgreachtomomentthisfromdrugthetakesittimethebetLet
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