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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Population The table shows the populations P (in thousands) of Lakewood, Colorado, from 2009 through 2013. (Source: U.S. Census Bureau)

Year 2009 2010 2011 2012 2013
Population, P 142 143 144 146 147

(a) Use the 2009 and 2010 data to find an exponential model P 1 .   L e t    t =   9 represent 2009.

(b) Use a graphing utility to find an exponential model P 2 for all of the data. L e t    t =   9 represent 2009.

(c) Use a graphing utility to plot the data and graph both models in the same viewing window. Compare the actual data with the estimates from the models. Which model is more accurate?

(a)

To determine

To calculate: The exponential model for the following data by using 2009 and 2010 data, assuming t=9 represents 2009,

Year 2009 2010 2011 2012 2013
Population, P 142 143 144 146 147
Explanation

Given Information:

The provided data is:

Year 2009 2010 2011 2012 2013
Population, P 142 143 144 146 147

Formula used:

Exponential growth and decay:

If the rate of change of a positive quantity y with respect to time is proportional to the amount of quantity present at any time t, that is dydt=ky, then y is given by the equation, y=Cekt, where C is the value of the quantity at time t=0 and k is the constant of proportionality.

If k>0 then there is exponential growth and when k<0 then there is exponential decay.

Calculation:

Consider the provided data:

Year 2009 2010 2011 2012 2013
Population, P 142 143 144 146 147

Observe that the population at time t=9 is 142 and the population at time t=10 is 143

Substitute 9 for t and 142 for y in exponential model y=Cekt,

142=Cek9Ce9k=142

Substitute 10 for t and 143 for y exponential model y=Cekt,

143=Cek10Ce10k=143

Take the ratio of the equations Ce9k=142 and Ce10k=143

(b)

To determine

The exponential model using graphing utility for the following data assuming t=9 represents 2009,

Year 2009 2010 2011 2012 2013
Population, P 142 143 144 146 147

(c)

To determine

To graph: The estimated exponential model calculated in part (a) and the exponential regression model calculated using Ti-83 graphing utility along with the data points where the provided table is:

Year 2009 2010 2011 2012 2013
Population, P 142 143 144 146 147

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P-107RECh-4 P-108RECh-4 P-109RECh-4 P-110RECh-4 P-111RECh-4 P-112RECh-4 P-113RECh-4 P-114RECh-4 P-115RECh-4 P-116RECh-4 P-117RECh-4 P-118RECh-4 P-119RECh-4 P-120RECh-4 P-121RECh-4 P-122RECh-4 P-123RECh-4 P-124RECh-4 P-125RECh-4 P-126RECh-4 P-127RECh-4 P-128RECh-4 P-129RECh-4 P-130RECh-4 P-131RECh-4 P-132RECh-4 P-1TYSCh-4 P-2TYSCh-4 P-3TYSCh-4 P-4TYSCh-4 P-5TYSCh-4 P-6TYSCh-4 P-7TYSCh-4 P-8TYSCh-4 P-9TYSCh-4 P-10TYSCh-4 P-11TYSCh-4 P-12TYSCh-4 P-13TYSCh-4 P-14TYSCh-4 P-15TYSCh-4 P-16TYSCh-4 P-17TYSCh-4 P-18TYSCh-4 P-19TYSCh-4 P-20TYSCh-4 P-21TYSCh-4 P-22TYSCh-4 P-23TYSCh-4 P-24TYSCh-4 P-25TYSCh-4 P-26TYSCh-4 P-27TYSCh-4 P-28TYSCh-4 P-29TYS

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