   Chapter 13.1, Problem 40E ### 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

# 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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