# Use the exponential growth model A=A_0e^{kt} to solve: In 2000, there were 110 million cellphone subscribers in the United States. By 2010, there were

Use the exponential growth model $A={A}_{0}{e}^{kt}$ to solve: In 2000, there were 110 million cellphone subscribers in the United States. By 2010, there were 303 million subscribers
a. Find the exponential function that models the data.
b. According to the model, in which year were there 400 million cellphone subscribers in the United States?
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Given Data:
The exponential growth model is $A={A}_{0}{e}^{kt}$
In 2000 the number of telephone users was 110 million.
In 2010 the number of telephone users was 303 million.
Taking 2000 as the base year (t=0) and A as the number of cellphone users then, boundary condition can be written as,
$t=0,A=110$ million
$t=10,A=303$ million
Substituting the boundary condition in the model equation,
101 million $={A}_{0}{e}^{k×0}$
${A}_{0}=110$ million
303 million ${A}_{0}{e}^{k×10}$
303 million = 110 million$×{e}^{k×10}$
${e}^{10k}=2.75$
$10k=\mathrm{ln}2.75$
$k=0.101325$
Thus, the required model equation is,
$A=110{e}^{0.101325t}$
The time after which the telephone users will be 400 million can be determined as,
$400=110{e}^{0.101325t}$
${e}^{0.101325t}=3.636$
$0.101325t=1.290$
$t=12.74$
Thus, the year in which the number of telephone users will be 400 million is
$2000+12.74=2012.74$
Thus, in the year 2013 the number of telephone users will be 400 million.