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Applications of Exponential Functions - Growth & Decay

• There are many applications of exponential functions in the areas of growth and decay.

Growth Model Decay Model

0k,eA)t(A tk0 0k,eA)t(A tk

0

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Applications of Exponential Functions - Growth & Decay

• Example 1:

Consider the model representing the amount of decay of carbon-14, where ...

t000121.00eA)t(A

t = time in years A0 = initial amount of carbon-14A(t) = amount of carbon-14 after t years

Assume that a bone originally had 20 grams of carbon-14 present. How many grams will be present 1000 years later?

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Applications of Exponential Functions - Growth & Decay

)1000)(000121.0(e20)1000(A

• Letting A0 = 20 and t = 1000 yields ...

... or approximately 17.72 grams of carbon-14 remaining.

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Applications of Exponential Functions - Growth & Decay

• Example 2:

Consider the modelt2e4001

000,60)t(N

t = time in weeks N(t) = is the number of people who have the flu in a

certain state t weeks after the initial outbreak.

where ...

Find the following:a) the number of people ill with the flu when the epidemic began.b) the number of people ill with the flu after 3 weeks.c) the total number of people with the flu at the end of the epidemic.

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Applications of Exponential Functions - Growth & Decay

a) Letting t = 0 represent the beginning of the epidemic, there were approximately 150 people ill with the flu initially.

• The most efficient way to answer the questions would be to use the TABLE feature on a graphing calculator. Type in the formula into y1, set TBLSET to 0, 1, auto, and then go to TABLE.

t2e4001

000,60)t(N

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Applications of Exponential Functions - Growth & Decay

c) To find the total number of people with the flu at the end of the epidemic, consider what value N(t) is approaching as the value of t increases. Scrolling downin the table yields ...

b) A value of t = 3 represents the number of people ill after 3 weeks, or approximately 30,128 people.

which suggests 60,000 people.

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Applications of Exponential Functions - Growth & Decay