54. U.S. households with cable/satellite TV Suppose therate of change of the percent P of U.S. households withcable/satellite TV can be modeled bydP46.20%3Ddtt + 5where t is the number of years past 1975.(a) Use integration and the data point for 2007 to find thefunction P(t) that models the percent of U.S. householdswith cable/satellite TV.(b) How well does the model from part (a) fit the data in thetable? Explain.(c) If the model remains valid, use it to predict the percentof U.S. households with cable/satellite TV in 2020.YearPercentYearPercent198022.6200067.8198546.2200585.7199059.0200783.8199565.7200989.7Source: Nielsen Media Research 55. Consumer price index The Social Security Administra-tion makes projections about the consumer price index(CPI) in order to understand the effects of inflation onSocial Security benefits and to plan for cost-of-livingincreases. Suppose the rate of change of the CPI can bemodeled with the functiondC= 3.087e0.03841dtdollars per year, where C is the consumer price indexand t is the number of years past 1990.(a) Does the model for the rate reflect the fact that theSocial Security Administration's data (actual andprojected for selected years from 1995 to 2070) inthe table show that the CPI is increasing? Explain.(b) Use integration and the table's data point for 2005to find the function that models the Social SecurityAdministration's CPI figures.(c) Find and interpret C(35) and C'(35).YearCPIYearCPI1995100.002035465.982000118.212040566.942005143.672045689.772010174.802050839.212015212.6720551021.022020258.7420601242.232025314.8020651511.362030383.0020701838.81Source: Social Security Administration

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54. U.S. households with cable/satellite TV Suppose the rate of change of the percent P of U.S. households with cable/satellite TV can be modeled by dP 46.20 %3D dt t+ 5 where t is the numb

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

Linear Correlation

A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.

Question

100%

54. U.S. households with cable/satellite TV Suppose the
rate of change of the percent P of U.S. households with
cable/satellite TV can be modeled by
dP
46.20
%3D
dt
t + 5
where t is the number of years past 1975.
(a) Use integration and the data point for 2007 to find the
function P(t) that models the percent of U.S. households
with cable/satellite TV.
(b) How well does the model from part (a) fit the data in the
table? Explain.
(c) If the model remains valid, use it to predict the percent
of U.S. households with cable/satellite TV in 2020.
Year
Percent
Year
Percent
1980
22.6
2000
67.8
1985
46.2
2005
85.7
1990
59.0
2007
83.8
1995
65.7
2009
89.7
Source: Nielsen Media Research

55. Consumer price index The Social Security Administra-
tion makes projections about the consumer price index
(CPI) in order to understand the effects of inflation on
Social Security benefits and to plan for cost-of-living
increases. Suppose the rate of change of the CPI can be
modeled with the function
dC
= 3.087e0.03841
dt
dollars per year, where C is the consumer price index
and t is the number of years past 1990.
(a) Does the model for the rate reflect the fact that the
Social Security Administration's data (actual and
projected for selected years from 1995 to 2070) in
the table show that the CPI is increasing? Explain.
(b) Use integration and the table's data point for 2005
to find the function that models the Social Security
Administration's CPI figures.
(c) Find and interpret C(35) and C'(35).
Year
CPI
Year
CPI
1995
100.00
2035
465.98
2000
118.21
2040
566.94
2005
143.67
2045
689.77
2010
174.80
2050
839.21
2015
212.67
2055
1021.02
2020
258.74
2060
1242.23
2025
314.80
2065
1511.36
2030
383.00
2070
1838.81
Source: Social Security Administration

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