4.2 Applications of Exponential Functions

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1 . Applications of Eponential Functions In this section ou will learn to: find eponential equations usin raphs solve eponential rowth and deca problems use loistic rowth models Eample : The raph of is the transformation of f ( ) =. Find the equation of the raph of. HINTS:. There are no stretches or shrinks.. Look at the eneral raph and asmptote to determine an reflections and/or vertical shifts.. Follow the point (0, ) on f throuh the transformations to help determine an vertical and/or horizontal shifts. 6 Eample : The raph of is the transformation of f ( ) Find the equation of the raph of. Eample : The raph of is the transformation of f ( ) Find the equation of the raph of. 6 Pae (Section.)

2 Eample : In 969, the world population was approimatel.6 billion, with a rowth rate of.7% per 0.07 ear. The function f ( ) =.6e describes the world population, f (), in billions, ears after 969. Use this function to estimate the world population in Eample : The eponential function millions, ears after 986. f ) ( ) = 8.(.0 models the population of Meico, f (), in (a) Without usin a calculator, substitute 0 for and find Meico s population in 986. (b) Estimate Meico s population, to the nearest million in the ear 000. (c) Estimate Meico s population, to the nearest million, this ear. Eample 6: One application of the natural eponential function involves Newton s Law of Coolin. This (law) formula models the temperature of an object as it cools down. For eample, when a pizza is removed from the oven and placed on the kitchen counter. The function model is T ( R 0 k ) = T + ( T T ) e, k <0 where T 0 = initial temperature of the object R T R = temperature of the room or surroundin area = number of minutes T() = temperature of the object minutes later k = coolin rate determined b the nature and phsical properties of the object A pizza is taken from a 0 deree oven and placed on the counter to cool. The temperature in the kitchen is 70 derees and the coolin rate for this tpe of pizza is k = -0.. (a) Use these values to obtain the equation model for T(). (b) Use the TABLE feature on our calculator to find the temperature of the pizza for each of the iven times. Minutes Temperature of the Pizza Pae (Section.)

3 Loistic Growth Models: Loistic rowth models situations when there are factors that limit the abilit to row or spread. From population rowth to the spread of disease, nothin on earth can ehibit eponential rowth indefinitel. Eventuall this rowth levels off and approaches a maimum level (which can be represented b a horizontal asmptote). Loistic rowth models are used in the stud of conservation biolo, learnin curves, spread of an epidemic or disease, carrin capacit, etc. The mathematical model for limited loistic rowth is iven c c b: f ( t) = or A =, where a, b, and c are constants, c > 0 and b > 0. + ae + ae As time increases ( t ), the epression ae and A. Therefore = c is a horizontal asmptote for the raph of the function. Thus c represents the limitin size. Eample7: A farmer wants to stock a private lake on his propert with catfish. A specialist studies the area and the depth of the lake, alon with other factors, and determines it can support a maimum population of approimatel 70 fish, with rowth modeled b the loistic function: 70 =,where f(t) ives the current population after t months. + e f ( t) 0. 07t (a) How man catfish did the farmer initiall put in the lake? (b) Based on this model, how man catfish were in the lake after 0 ears? After 0 ears? (c) What is the limitin size of the catfish population? (d) What is the horizontal asmptote for this function? (e) Sketch a raph of this function. Number of Catfish Months Pae (Section.)

4 . Homework Problems. Find the equation of each eponential function, (), whose raph is shown. Each raph involves one or more transformation of the raph of f ( ) =. (a) (b) (c) Find the equation of each eponential function, (), whose raph is shown. Each raph involves one or more transformation of the raph of f ( ) (a) (b) (c) In 970, the U. S. population was approimatel 0. million, with a rowth rate of.% per ear. 0.0 The function f ( ) = 0.e describes the U. S. population, f (), in millions, ears after 970. Use this function to estimate the U. S. population in the ear 0.. A common bacterium with an initial population of 000 triples ever da. This is modeled b the t formula P ( t) = 000(), where P(t) is the total population after t das. Find the total population after (a) hours (b) da (c) ½ das (d) das. Assumin the rate of inflation is % per ear, the predicted price of an item can be modeled b the t function P t) = P (.0), where P 0 represents the initial price of the item and t is in ears. ( 0 (a) Based on this information, what will the price of a new car be in the ear 0, if it cost $0,000 in the ear 000? (b) Estimate the price of a allon of milk be in the ear 0, if it cost $.9 in the ear 000? Round our estimate to the nearest cent. Pae (Section.)

5 6. The 986 eplosion at the Chernobl nuclear power plant in the former Soviet Union sent about kilorams of radioactive cesium-7 into the atmosphere. The function f ( ) = 000(0.) describes the amount, f (), in kilorams, of cesium-7 remainin in Chernobl ears after 986. If even 00 kilorams of cesium-7 remain in Chernobl s atmosphere, the area is considered unsafe for human habitation. Find f (60) and determine if Chernobl will be safe for human habitation b , The loistic rowth function f ( t) = describes the number of people, f (t), who have t + 000e become ill with influenza t weeks after its initial outbreak in a particular communit. (a) How man people became ill with the flu when the epidemic bean? (b) How man people were ill b the end of the fifth week? (c) What is the limitin size of the population that becomes ill? The loistic rowth function P( ) = models the percentae, P () of Americans who e are ears old with some coronar heart disease. (a) What percentae of 0-ear-olds have some coronar heart disease? (b) What percentae of 80-ear-olds have some coronar heart disease?. Homework Answers:. (a) ( ) = ; (b) ( ) = ; (c) ( ) = +. (a) ( ) = e ; (b) ( ) = e ; (c) ( ) = e. about.7 million. (a) about 7; (b) 000; (c) about 96; (d) (a) about $,97.; (b) $ ; no 7. (a) about 0 people; (b) about 88 people; (c) 00,000 people 8. (a) about.7%; (b) about 88.6% Pae (Section.)