P t) = P model. The constant is called b The growth rate ...

1 A function which models exponential growth or decay can be written in either the formP(t) =P0btorP(t) = either form,P0represents the initial formP(t) =P0ektis sometimes called thecontinuous exponentialmodel. The constantkis calledthecontinuous growth (or decay) rate. In the formP(t) =P0bt, thegrowth rateisr=b 1. The constantbis sometimes called the growth growth rate and growth factor are not the is a simple matter to change from one model to the other. If we are givenP(t) =P0ekt, and wantto write it in the formP(t) =P0bt, all that is needed is to note thatP(t) =P0ekt=P0(ek)t, so if we letb=ek, we have the desired form. If we want to switch fromP(t) =P0bttoP(t) =P0ekt, it again is just amatter of noting thatb=ek, and solving forkin this case. That is,k= for solving problems:Given a rate of growth or decayr, Ifris given as a constant rate of change (some fixed quantity per unit), then the equation is linear( +rt) Ifris given as a percentage increase or decrease, then the equation is exponential if the rate of growth isr, then use the modelP(t) =P0bt, whereb= 1 +r if the rate of decay isr, then use the modelP(t) =P0bt, whereb= 1 r if thecontinuousrate of growth isk, then use the modelP(t) =P ekt, withkpositive if thecontinuousrate of decay isk, then use the modelP(t) =P ekt, withknegativeThus, the formP(t) =P0ektshould be used if the rate of growth or decay is stated as :A trash dumpster starts with 5 pounds of garbage.

2 Write a function which represents the amount ofgarbage in the dumpster aftertdays given the following ratesa) The amount of garbage increases by 3 lbs per day Since the rate is stated as a constant 3 pounds per day, the equation is linear. So, the modelisQ(t) = 5 + 3t, whereQrepresents the amount of trash, andtis measured in ) The amount of garbage increases by 3% per day Since the rate is now given as percentage increase, we need to use the exponential modelP(t) =P0bt. Since the growth rate isr=.03, the base of our model should beb= 1 +. , if we again letQrepresent the amount of trash, we haveQ(t) = 5( )t. Note that the growthrateis .03, and the growthfactoris ) The amount of garbage increases continuously by 3% per day Since the rate is given as acontinuouspercentage increase, we need to use the exponentialmodelP(t) =P0ekt.

3 We havek=.03, so our model isQ(t) = :Suppose the population of ants in a colony grows by per ) Determine a model which represents the population of the colony aftertmonths. What is thegrowth rate? What is thecontinuous growth rate? The model is simplyP(t) =P0( ) growth rate isr=.042 (or ).In order to find the continuous growth rate, we need to convert the model to the formP(t) =P0ekt. So, we need to solve forkin =ek. Taking the natural log of both sides,we getk= ln( ) .04114. Thus the continuous growth rate is approximately .04114 (orabout ).b) Suppose instead that the population of ants grows at a continuous rate of a model which represents the population of the colony aftertmonths. What is thegrowth rate? What is the continuous growth rate? In this case the model isP(t) = find the growth rate, we convert to the formP(t) =P0bt.

4 So,b= Thusthe growth factor is about , and the growth rate is approximately .04289 (or ).The continuous growth rate is the stated :A container with 1 liter of a liquid is placed in a warm, arid environment. The liquid evaporates ata rate of per day. Write a function which represents the amount of liquid (in milliliters) in thecontainer aftertdays. Since the amount of liquid present is decreasing, we have an example of exponential decay. The rate of decay is , so the base of our exponential isb= 1 .023 =.977. (If we like,we could also think of the rate of decay, as a rate of growth of , which yields the sameresult since thenb= 1 +r= 1 + ( .023) =.997). In any case, our exponential model isQ(t) = 1000(.977)t(the initial amount is 1000, since our function is supposed to representthe amount inmilliliters).

5 EXERCISES1. Write a model to represent each of the following.(a) The amount of a radioactive substance present afterthours if there are initially 500 mg and thehalf-life is 9 hours.(b) The value of a car aftertyears if the initial price is $26,000 and the car depreciates at a rate of11% per year.(c) The value of a car aftertyears if the initial price is $33,000 and the car depreciates at a rate of$3700 per year.(d) The population of a city in yeartif the population in 2005 is 125,000 and the city is growing ata rate of per Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the originalamount to answer the following.(a) Write an equation which represents the amount of carbon-14 present aftertyears. What does thebase of your equation tell you (in practical terms)?

6 (b) In 1947, earthenware jars containing what are now known as the Dead Sea Scrolls were indicated that the scroll wrappings contained 76% of their original carbon-14. Estimatethe current age of the Dead Sea Suppose that the fruit fly population in a small habitat aftertdays is given by the logistics equationP(t) =2301 + .37t(a) How many fruit flies were initially placed in the habitat?(b) What is the carrying capacity of the habitat? [That is, ast what happens toP(t)?] Whatfeature is this on the graph?(c) How many fruit flies are present on day 4?(d) When will the population of fruit flies be 180?4. It is predicted that the population of a certain town in 2030 will be double the population in 2005.(a) Determine the annual growth rate.(b) Determine the monthly growth rate.

7 (c) Determine the continuous growth rate.