K.ee# growth of the population 's. Ky! LECTURE: 3-8EXPONENTIAL GROWTH AND DECAY. yco2=c. dd = C. ekt. decreasing. population. Population at time to
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1 LECTURE: 38EXPONENTIAL GROWTH AND DECAY In many natural phenomena a quantity grows or decays at a rate proportional to their size Suppose y f(t is the number of individuals in a population at time t Given an unlimited environment adequate nutrition and immunity to disease it is reasonable to assume that the rate of growth is proportional to the population That is constant of f proportionality f 0 (t dy dt Example 1: Show that the equation y Ce kt is a solution to the differential equation dy dt ky (a Explain in words what it means for y Ce kt to be a solution of the given differential equation When you check Trak Ky of this in dy/dt Ky Population growth it will work ' ' (b Show that y Ce kt is a solution to the differential equation dy dt ky dd C ekt * at K Kee# at mm Ky! Theorem: The only solutions of the differential equation dy/dt ky are exponential functions of the form y(t Ce kt where C y(0 Explain why C y(0 y ( o ceh0 ' This means that C is the initial yco2c j Population at time to What does the constant k mean in this equation? What does the sign of k tell you about the growth of your population? K is the rate of growth of the population 's H K > 0 the population is growing If KCO the population is decreasing UAF Calculus I 1 38 Exponential Growth and Decay
2 1h40 Example 1: A bacteria culture initially contains 10 cells and grows at a rate proportional to its size After an hour the population has increased to 400 (a Find an expression for the number of bacteria after t hours 10 f ( SO f Lt Cekt fgof ;Ee flt 10 ekt nonfattening f( e ek K In 40 k 1 (b Find the number of bacteria after 3 hours f ( (c Find the rate of growth after 3 hours ( 10 pm 40T 1h40 I ( (d When will the population reach 1000? 640O0Obaoteriaf' 2}6O 8285O6bacteri% jn40t In 40 t 100 e In 100 1h40 t t II 1h40 T UAF Calculus I 2 38 Exponential Growth and Decay
3 Example 2: Let y Ce kt be the number of flies at time t where t is measured in days Suppose there are 100 flies after the second day and a 300 flies after the fourth day Assuming the growth rate is proportional to the population size find a model for this population s growth When will the population of flies be 10000? ( 2100 and ( ce2k C 10%#l Example 3: The halflife of cesium137 is 30 years The 1986 explosion at Chernobyl sent about 1000 kg of radioactive cesium137 into the atmosphere (a Find the mass that remains after t years y zgeltzln kz#l LY#tY2s@ e4k Ce EZK C e2k e2k C2@ 4000 zt Cef yt m4 In 4 2k In ( 4000 In Zt (b If even 100 kg remains in Chernobyl s atmosphere the area is considered unsafe for human habitation Determine when Chernobyl will be safe UAF Calculus I 3 38 Exponential Growth and Decay t 2 In t In 2 + lnn4g _ k 30k Yz e mh1ooo(yz#3 _ mlt C ekt MCYZ 30k K Yzo lnllk mlt 10000kt mlt Know 30 mbo mL t 1000 ( ( Yzj 2s%t '3ot (1/2*30 looo(yz5'3 # 10(1/2*30 mulatto Mato (B In z±o1nhz thn#n(yz~~ a p 100 SO years 2086T after 198$
4 Example 4: A sample of radioactive tritium3 decayed to 95% of its original amount after a year (a What is the halflife of tritium3? tec#nsgynyoae5etfisnyisstly@95c h 1h10 MLO 5 so 095C c( 05 h h In 05/1 no gs 095 ( 05k (b How long would it take the sample to decay to 10% of its original amount? 01C C ( O NO D g 5 +1%513 t In Tss 53 In 105 t44891years& Example 5: Scientists can determine the age of ancient objects by the method of radiocarbon dating The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon 14 C with a half life of about 5730 years Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14 C through food chains When a plant or animal dies it stops replacing its carbon and the amount of 14 C begins to decrease through radioactive decay Therefore the level of radioactivity must also decay exponentially A parchment fragment was discovered that had about 74% as much 14 C radioactivity as does the plant material on earth today Estimate the age of the parchment t 5730 In 1074 flt C (1/2%730 1 a : when is fat 074C? C C ( * ( O g In 1074 In In In 105 UAF Calculus I 4 38 Exponential Growth and Decay
5 ' Newton s Law of Cooling The rate of cooling ( or warming of an object IS proportional the temperature difference between the object : surroundings date K ( T Ts e temp temp of surroundings Example 6: When a cold drink is taken from a refrigerator its temperature is 40 F After 25 minutes in a 70 F room its temperature has increased to 52 F and its (a What is the temperature of the drink after 50 minutes? not quite lg ky close let YT Ts this works Have Y T 70 here ; yco Thus ylt 30 ekt also after # 25 min drink is 52 F So YC and ; 5ytEo so se %5 (b When will its temperature reach 60 Mas F? 25m (41 5 t In:* T(tz7OzoeYzsml95_K e In (9/15/25 Tl5#s92 F] 5 t87 689mi_ In ( ' e 25 In ( 46 In ( 415 ' 1251hL 9115 t 425 In ( 415 t t Yzs In ( 9115 t (c What happens to the temperature of the drink as t!1? Is this expected? Itmgatttt L ;ma ( 20 fitz( 70 Zoe 06385T 3%0 6385T 7O@ as + UAF Calculus I 5 38 Exponential Growth and Decay Its 70 F or the ambient room temp
33. The gas law for an ideal gas at absolute temperature T (in. 34. In a fish farm, a population of fish is introduced into a pond
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