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Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

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BuyFindarrow_forward

Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

Social Security beneficiaries The table gives the number of millions of Social Security beneficiaries (actual and projected) for selected years from 1950 through 2030.

Year Number of Beneficiaries (millions) Year Number of Beneficiaries (millions)
1950 2.9 2000 44.8
1960 14.3 2010 53.3
1970 25.2 2020 68.8
1980 35.1 2030 82.7
1990 39.5

Source: Social Security Trustees Report

With B(t) representing the number of beneficiaries (in millions) t years past 1950, these data can be modeled by the function

B ( t )   = ( 0.01 t   +   3 ) ( 0.0238 t 2   9.79 t   +   3100 )   9290

(a) Find the function that gives the instantaneous rate of change of the number of beneficiaries.

(b) Find and interpret the instantaneous rate of change in 2020.

(c) Use the data to determine which of the average rates of change (from 2010 to 2020, from 2020 to 2030, or from 2010 to 2030) best approximates the instantaneous rate from part (b).

(a)

To determine

To calculate: The function for the instantaneous rate of change of the number of beneficiaries, if the function for the number of beneficiaries (in millions) t years past 1950 model as B(t)=(0.01t+3)(0.0238t29.79t+3100)9290. The data that gives the number of Social Security beneficiaries (actual and projected) for selected years from 1950 through 2030 for which the function is modeled is tabulated below

Year Number of Beneficiaries (millions) Year Number of Beneficiaries (millions)
1950 2.9 2000 44.8
1960 14.3 2010 53.3
1970 25.2 2020 68.8
1980 35.1 2030 82.7
1990 39.5
Explanation

Given Information:

The function for the number of beneficiaries (in millions) t years past 1950 model as B(t)=(0.01t+3)(0.0238t29.79t+3100)9290.

The table that gives the number of Social Security beneficiaries (actual and projected) for selected years from 1950 through 2030 for which the function is modeled is,

Year Number of Beneficiaries (millions) Year Number of Beneficiaries (millions)
1950 2.9 2000 44.8
1960 14.3 2010 53.3
1970 25.2 2020 68.8
1980 35.1 2030 82.7
1990 39.5

Formula Used:

The product rule for the derivative of the two function, ddx(fg)=fdgdx+gdfdx.

The sum or difference rule of derivate of functions, ddx[u(x)±v(x)]=ddxu(x)±ddxv(x).

The simple power rule of derivative ddx(xn)=nxn1.

Calculation:

Consider the model function B(t)=(0.01t+3)(0.0238t29.79t+3100)9290.

Differentiate the provided model function,

dB(t)dt=ddt((0.01t+3)(0.0238t29.79t+3100)9290)

Use the sum or difference rule of derivative, ddx[u(x)±v(x)]=ddxu(x)±ddxv(x).

dB(t)dt=ddt(0.01t+3)(0.0238t29.79t+3100)

Use the product rule for the derivative of the two function f(x) and g(x) is, ddx(fg)=fdgdx+gdfdx

(b)

To determine

To calculate: The instantaneous rate of change of the number of beneficiaries in 2020 and its interpretation, if the function for the number of beneficiaries (in millions) t years past 1950 model as B(t)=(0.01t+3)(0.0238t29.79t+3100)9290.

The data that gives the number of Social Security beneficiaries (actual and projected) for selected years from 1950 through 2030 for which the function is modeled is tabulated below

Year Number of Beneficiaries (millions) Year Number of Beneficiaries (millions)
1950 2.9 2000 44.8
1960 14.3 2010 53.3
1970 25.2 2020 68.8
1980 35.1 2030 82.7
1990 39.5

(c)

To determine

The time range among (from 2010 to 2020, from 2020 to 2030 orfrom 2010 to 2030) for which the average rate of change is the best approximates of the instantaneous rate of change in 2020, if the function for the number of beneficiaries (in millions) t years past 1950 model as B(t)=(0.01t+3)(0.0238t29.79t+3100)9290.

The data that gives the number of Social Security beneficiaries (actual and projected) for selected years from 1950 through 2030 for which the function is modeled is tabulated below

Year Number of Beneficiaries (millions) Year Number of Beneficiaries (millions)
1950 2.9 2000 44.8
1960 14.3 2010 53.3
1970 25.2 2020 68.8
1980 35.1 2030 82.7
1990 39.5

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