The number N of US cellular phone subscribers (in millions) is shown in the table. (Midyear estimates are given.)

t 1996 1998 2000 2002 2004 2006
N 44 69 109 141 182 233

(a) Find the average rate of cell phone growth between the following years. In each case, include the units. (Round your answers to one decimal place.)
(i) from 1996 to 2004
million phones/yr

(ii) from 2002 to 2004
million phones/yr

(iii) from 2000 to 2002
million phone/yr

(b) Estimate the instantaneous rate of growth in 2002 by taking the average of the two average rates of change in (ii) and (iii). What are its units?
______million phones/yr

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(a)

(i) To find the average rate of cell phone growth from 1996 to 2004, we need to calculate the change in the number of subscribers divided by the number of years.

Change in subscribers = 182 million - 44 million = 138 million
Number of years = 2004 - 1996 = 8 years

Average rate of cell phone growth from 1996 to 2004 = Change in subscribers / Number of years = 138 million / 8 years = 17.3 million phones/yr

(ii) To find the average rate of cell phone growth from 2002 to 2004, we need to calculate the change in the number of subscribers divided by the number of years.

Change in subscribers = 182 million - 141 million = 41 million
Number of years = 2004 - 2002 = 2 years

Average rate of cell phone growth from 2002 to 2004 = Change in subscribers / Number of years = 41 million / 2 years = 20.5 million phones/yr

(iii) To find the average rate of cell phone growth from 2000 to 2002, we need to calculate the change in the number of subscribers divided by the number of years.

Change in subscribers = 141 million - 109 million = 32 million
Number of years = 2002 - 2000 = 2 years

Average rate of cell phone growth from 2000 to 2002 = Change in subscribers / Number of years = 32 million / 2 years = 16 million phones/yr

(b) To estimate the instantaneous rate of growth in 2002, we take the average of the two average rates of change in (ii) and (iii).

Instantaneous rate of growth in 2002 = (Average rate from 2002 to 2004 + Average rate from 2000 to 2002) / 2
= (20.5 million phones/yr + 16 million phones/yr) / 2 = 18.3 million phones/yr

The units for the instantaneous rate of growth in 2002 are million phones/yr.

To find the average rate of cell phone growth between two years, you need to calculate the change in the number of subscribers and then divide it by the number of years.

(a)

(i) From 1996 to 2004:

To find the change in the number of subscribers, subtract the number of subscribers in 1996 from the number of subscribers in 2004:
Change = 182 million - 44 million = 138 million

To find the number of years, subtract 1996 from 2004:
Years = 2004 - 1996 = 8 years

Average rate of cell phone growth = Change / Years = 138 million / 8 years ≈ 17.3 million phones/yr

(ii) From 2002 to 2004:

Change = 182 million - 141 million = 41 million
Years = 2004 - 2002 = 2 years

Average rate of cell phone growth = Change / Years = 41 million / 2 years = 20.5 million phones/yr

(iii) From 2000 to 2002:

Change = 141 million - 109 million = 32 million
Years = 2002 - 2000 = 2 years

Average rate of cell phone growth = Change / Years = 32 million / 2 years = 16 million phones/yr

(b)

To estimate the instantaneous rate of growth in 2002, take the average of the two average rates of change in (ii) and (iii):

Instantaneous rate of growth in 2002 = (Average rate from 2002 to 2004 + Average rate from 2000 to 2002) / 2
= (20.5 million phones/yr + 16 million phones/yr) / 2
= 18.25 million phones/yr

The units for the instantaneous rate of growth in 2002 are million phones/yr.

(I) 18.8

(II) 16
(III) 20
(b) 18