   Chapter 11, Problem 19T ### Mathematical Applications for the ...

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

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

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

# Prices for goods and services tend to rise over time, and this results in the erosion of purchasing power. With the 2012 dollar as a reference, a purchasing power of 0.921 for a certain year means that in that year, a dollar will purchase 92.1% of the goods and services that could be purchased for \$1 in 2012. Using Social Security Administration data for selected years from 2012 and projected to 2050, the purchasing power of a 2012 dollar can be modeled by the function P ( t ) =   1.078 ( 0.9732 t ) where t is the number of years past 2010.(a) Find the function that models the rate of change of the purchasing power of a 2012 dollar.(b) Find and interpret P ( 18 )  and  P ' ( 18 ) .

(a)

To determine

To calculate: The rate of change of the purchasing power of a 2012 dollar. If the purchasing power of a 2012 dollar can be modeled by the function P(t)=1.078(0.9732t). Where t is the number of years past 2010.

Explanation

Given Information:

The purchasing power of a 2012 dollar can be modeled by the function as;

P(t)=1.078(0.9732t)

Where t is the number of years past 2010.

Formula used:

The derivative of function f(x)=e(ax+b) is given by;

f(x)=e(ax+b)(a)

The derivative of function f(x)=xn is given by;

f(x)=nxn1

If y=ax, with a>0, a1, then;

dydx=axlna

Calculation:

The rate of change of the purchasing power of a 2012 dollar is given by the derivative as;

P(t)=ddt[1.078(0

(b)

To determine

To calculate: The P(18), P(18) and interpret if the purchasing power of a 2012 dollar can be modeled by the function P(t)=1.078(0.9732t). Where t is the number of years past 2010.

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