Question

The ratio of working-age population to the elderly in the United States (including projections after 2000) is given by the function below, with t = 0 corresponding to the beginning of 1995.†
$f(t) = {{4.1 if 0 \leq t \ \textless \ 5}, {-0.03t + 4.25 if 5 \leq t 15}, {-0.075t + 4.925 if 15 \leq t \leq 35}$
(a) Sketch the graph of the function f.
(b) What was the ratio at the beginning of 2006? At the beginning of 2014?
2006
2014
(c) Over what years is the ratio constant?
[1995, 2000]
[2010, 2030]
[0, 5]
[5, 15]
[2000, 2010]
(d) Over what years is the decline of the ratio greatest?
[15, 35]
[2010, 2030]
[1995, 2000]
[2000, 2010]
[5, 15]

Answer 1

Answer:

a) Sketch the graph of the function f. (it is in the attached file)

b) What was the ratio at the beginning of 2006? At the beginning of 2014?

For 2006 the ratio is 3.92

For 2014 the ratio is 3.5

c) Over what years is the ratio constant?

[1995, 2000]

d) Over what years is the decline of the ratio greatest?

[2010, 2030]

Step-by-step explanation:

b) We first need to know in between which function 2006 falls into, so if we start in t=0=1995, then 2006-1995=11, t=11

t=11 fall into:

$f(x)=-0.03t+4.25$

$f(11)=-0.03(11)+4.25=3.92$

For 2006 the ratio is 3.92

2014:

Same process that 2006, t=0=1995, then 2014-1995=19, t=19

t=19fall into:

$f(x)=-0.075t+4.925$

$f(19)=-0.075(11)+4.925=3.5$

For 2014 the ratio is 3.5

c) The ratio is only constant in the first section of the graph 0≤t<5, since 4.1 is constant. Following the same process for the years in star (b) we have t=0, 1995+0=1995, t=5, 1995+5=2000.

The ratio will be constant between [1995, 2000]

d) For the greatest decline we need to compare slopes. From the line equation we have:

$y(x)=mx+b$ where m is the slope and b is the point the line intersects with the y axis. Here we have:

$f(x)=-0.03t+4.25$  for 5≤t<15

$f(x)=-0.075t+4.925$  for 15≤t≤35

So:

m=-0.03 for 5≤t<15

m=-0.075 for 15≤t≤35

If we are measuring the steepness of the decline, we have to compare:

|-0.03| and |-0.075| or 0.03 and 0.075, easily finding that 0.075>0.03

And doing the sames process for the years in question (c):

t=15, 1995+15=2010, and t=35, 1995+35=2030

This means that the biggest decline is between the years:

[2010, 2030]