Chapter 9.7, Problem 45E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Per capita expenditures for U.S. health care The dollars spent per person per year for health care (projected to 2018) are shown in the table.These data can be modeled by y = 4.38 ( x − 10 ) 2 + 78 ( x − 10 ) + 1430 0.0029 x + 0.25 where x is the number of years past 1990 and y is the per capita expenditures for health care.(a) Find the instantaneous rate of change of per capita health care expenditures in 2005 and 2015.(b) Interpret the rate of change for 2015 found in part (a).(c) Use the data to find the average rate of change of per capita health care expenditures from 2004 to 2006. How well does this approximate the instantaneous rate of change in 2005? Year $per Person Year$ per Person 2000 4789 2010 8465 2002 5563 2012 9275 2004 6331 2014 10,289 2006 7091 2016 11,520 2008 7826 2018 12,994 Source: U.S. Medicare and Medicaid Services

(a)

To determine

To calculate: The instantaneous rate of change of per capita health care expenditures for 2005 and 2015 for model function y=4.38(x10)2+78(x10)+14300.0029x+0.25.

Where x is the number of years after 1990 and y is the per capita expenditure for health care.

Explanation

Given Information:

The data given in the table

 Year $per person Year$ per person 2000 4789 2012 9275 2002 5563 2014 10,289 2004 6331 2016 11,520 2006 7091 2018 12,994 2008 7426 2010 8465

The model function for table is given bellows,

y=4.38(x10)2+78(x10)+14300.0029x+0.25

where x is the number of years after 1990 and y is the per capita expenditure for health care.

Formula used:

The rate of change of a function is given by its derivative. If a function is such that it is the ratio of two differentiable functions, then its derivative is calculated by the formula,

ddx[f(x)g(x)]=g(x)f(x)f(x)g(x)(g(x))2

Calculation:

Consider the model function,

y=4

(b)

To determine

To calculate: The impact of the instantaneous rate of change of per capita health care expenditures for 2015 when the data given in the table.

 Year $per person Year$ per person 2000 4789 2012 9275 2002 5563 2014 10,289 2004 6331 2016 11,520 2006 7091 2018 12,994 2008 7426 2010 8465

is modeled as

y=4.38(x10)2+78(x10)+14300.0029x+0.25

where x is the number of years after 1990 and y is the per capita expenditure for health care.

(c)

To determine

To calculate: The average rate of change of per capita health care expenditures from 2018 to 2015 when the data given in the table

is modeled as

y=4.38(x10)2+78(x10)+14300.0029x+0.25

where x is the number of years after 1990 and y is the per capita expenditure for health care.

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