(a) Explanation Given Information: The following table shows the total national expenditures for health (in billions of dollars) for selected years from 2002 and projected to 2024. Assume that these data can be modeled with the function A ( t ) = 333 ( 0.1 t + 1 ) 3 − 1830 ( 0.1 t + 1 ) 2 + 4510 ( 0.1 t + 1 ) − 1750 where A ( t ) is in billions of dollars and t is the number of years after 2000. Year Amount Year Amount 2002 1602 2014 3080 2004 1855 2016 3403 2006 2113 2018 3786 2008 2414 2020 4274 2010 2604 2022 4825 2012 2817 2024 5425 Formula used: The power rule of differentiation, d d x x n = n x n − 1 Calculation: Consider the provided function, A ( t ) = 333 ( 0.1 t + 1 ) 3 − 1830 ( 0.1 t + 1 ) 2 + 4510 ( 0.1 t + 1 ) − 1750 The instantaneous rate of change of total national health care expenditures is given by A ' ( t ) . Consider ( 0.1 t + 1 ) to be u , A ( t ) = 333 u 3 − 1830 u 2 + 4510 u − 1750 Differentiate both sides with respect to t , d A d t = d d t ( 333 u 3 − 1830 u 2 + 4510 u − 1750 ) Simplify using the power rule, d A d t = d d t ( 333 u 3 ) − d d t ( 1830 u 2 ) + d d t ( 4510 u ) − d d t ( 1750 ) = 333 d d t u 3 (b) To determine To calculate: The instantaneous rate of change of total national health care expenditures in 2012 and 2020 and interpret it. The model function A ( t ) = 333 ( 0.1 t + 1 ) 3 − 1830 ( 0.1 t + 1 ) 2 + 4510 ( 0.1 t + 1 ) − 1750 where A ( t ) is in billions of dollars and t is the number of years after 2000. (c) To determine To calculate: The average rate of change from 2018 to 2020 that approximates the instantaneous rate of change for 2020. The model function that these data can be modeled with the function A ( t ) = 333 ( 0.1 t + 1 ) 3 − 1830 ( 0.1 t + 1 ) 2 + 4510 ( 0.1 t + 1 ) − 1750 .

Mathematical Applications for the Management, Life, and Social Sciences

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ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 9.6, Problem 51E

(a)

To determine

To calculate: The instantaneous rate of change of function A(t)=333(0.1t+1)3−1830(0.1t+1)2+4510(0.1t+1)−1750, of total national health care expenditures.

(b)

To determine

To calculate: The instantaneous rate of change of total national health care expenditures in 2012 and 2020 and interpret it. The model function A(t)=333(0.1t+1)3−1830(0.1t+1)2+4510(0.1t+1)−1750 where A(t) is in billions of dollars and t is the number of years after 2000.

(c)

To determine

To calculate: The average rate of change from 2018 to 2020 that approximates the instantaneous rate of change for 2020. The model function that these data can be modeled with the function A(t)=333(0.1t+1)3−1830(0.1t+1)2+4510(0.1t+1)−1750.

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Chapter 9.6, Problem 50E

Chapter 9.6, Problem 52E

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Chapter 9 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 9.1 - 1. Can exist if is undefined? Ch. 9.1 - 2. Does exist if ? Ch. 9.1 - 3. Does Ch. 9.1 - 4. If , does exist? Ch. 9.1 - 5. Let (a) Does (b) Find . Ch. 9.1 - 6. Evaluate the following limits (if they...Ch. 9.1 - Prob. 7CP Ch. 9.1 - Prob. 8CP Ch. 9.1 - Prob. 9CP Ch. 9.1 - Prob. 10CP

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The instantaneous rate of change of function A ( t ) = 333 ( 0.1 t + 1 ) 3 − 1830 ( 0.1 t + 1 ) 2 + 4510 ( 0.1 t + 1 ) − 1750 , of total national health care expenditures.

Solution Summary: The author explains how the instantaneous rate of change of total national health care expenditures is given by A'(t).

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To calculate: The instantaneous rate of change of function A(t)=333(0.1t+1)3−1830(0.1t+1)2+4510(0.1t+1)−1750, of total national health care expenditures.

To calculate: The instantaneous rate of change of total national health care expenditures in 2012 and 2020 and interpret it. The model function A(t)=333(0.1t+1)3−1830(0.1t+1)2+4510(0.1t+1)−1750 where A(t) is in billions of dollars and t is the number of years after 2000.

To calculate: The average rate of change from 2018 to 2020 that approximates the instantaneous rate of change for 2020. The model function that these data can be modeled with the function A(t)=333(0.1t+1)3−1830(0.1t+1)2+4510(0.1t+1)−1750.

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Chapter 9 Solutions