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Mathematical Applications for the ...

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

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BuyFindarrow_forward

Mathematical Applications for the ...

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

Per capita income The per capita personal income (in dollars per year) in the United States for selected years from 1960 and projected to 2018 can be modeled by

I ( t )   = 13.93 t 2 +   136.8 t +   1971

where t is the number of years past 1960 (Source: U.S. Bureau of Labor Statistics). Use n = 10 equal subdivisions with right-hand endpoints to approximate the area under the graph of I(t) between t = 50 and t = 55. What does this area represent?

To determine

To calculate: The area under the graph of I(t)=13.93t2+136.8t+1971 between t=50 and t=55 using n=10 equal subdivisions measured at the right-handend points and what does this area represent where the function I(t)=13.93t2+136.8t+1971 represents per capita personal disposable income.

Explanation

Given Information:

The provided function is I(t)=13.93t2+136.8t+1971.

Formula used:

The base of the rectangles to approximate the area is ban where the interval [a,b] on which function is defined is divided into n subintervals.

The height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

The area of a rectangle is base×height.

The approximated area under the curve is sum of the areas of each rectangle.

Calculation:

Consider the function,

I(t)=13.93t2+136.8t+1971

The domain on which function is defined is [50,55].

Divide the interval [50,55] into n=10 subintervals where the function is evaluated at right end points of the subintervals.

The base of the rectangles to approximate the area is ban where the interval [a,b] on which function is defined is divided into n subintervals.

Since, the function is defined from t=50 to t=55. So, a=50 and b=55.

Thus,

Base of each rectangle=555010=510=0.5

Recall that the height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

Record these values in a table.

Rectangle Base Right endpoint Height Area=base×height
[50,50.5] 0.5 x1=50.5 I(50.5)=13.93(50.5)2+136.8(50.5)+1971=44404.3825 22202.19125
[50.5,51] 0.5 x2=51 I(51)=13.93(51)2+136.8(51)+1971=45179.73 22589.865
[51,51.5] 0.5 x3=51.5 I(51.5)=13.93(51.5)2+136.8(51.5)+1971=45962.0425 22981.02125
[51.5,52] 0.5 x4=52 I(52)=13

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Chapter 13 Solutions

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Sect-13.1 P-9ESect-13.1 P-10ESect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.2 P-1CPSect-13.2 P-2CPSect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 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