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Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640

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BuyFindarrow_forward

Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem

Working-Age Population The ratio of working-age population to the elderly in the United States (including projections after 2000) is given by

f ( t ) = { 4.1 if  0 t < 5 0.03 t + 4.25 if  5 t < 15 0.075 t + 4.925 if  15 t < 35

with t = 0 corresponding to the beginning of 1995.

  1. a. Sketch the graph of f.
  2. b. What was the ratio at the beginning of 2005? What will the ratio be at the beginning of 2020?
  3. c. Over what years is the ratio constant?
  4. d. Over what years is the decline of the ratio greatest?

Source: U.S. Census Bureau.

(a)

To determine

To sketch: The graph of the given function.

Explanation

Given:

The ratio of working-age population to the elderly in the United States (including projections after 2000) is given by the function as follows.

f(t)={4.1if0t<50.03t+4.25if5<t150.075t+4.925if15t35

Where t is measured in years, with t=0 corresponds to 1995.

Calculations:

Equate f(t) to 0 for interval 5t<15 to evaluate the zeros of f(t) .

0.03t+4.25=00.03t=4.25(Add0.03tbothsides)t=4.250.03(Dividebothsidesby0.03)t=141.6

The zero of f  in the interval 5t<15 is 141.6.

Substitute 0 for t and calculate y-intercept of f(t) as follows.

f(t)=0.03(0)+4.25=4.25

Substitute different values of t for the additional value of f(t) as shown below in the table 1.

t f(t)
04.25
14.22
24.19
34.16

Table 1

Equate f(t) to 0 for interval 15<t35 to evaluate the zeros of f(t)

(b)

To determine
The ratio at the beginning of 2005 and 2020.

(c)

To determine
The year of constant ratio.

(d)

To determine
The year when decline of ratio is greatest.

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