Respuesta :
For this case we have an exponential function of the form:
[tex]y = A (b) ^ x[/tex]
Where,
- A: initial number of subscribers
- b: rate of change
- x: number of years
Part A:
The equation modeling the problem is given by:
[tex]y = 233 (1.06) ^ x[/tex]
For the year 2008 we have x = 2.
Substituting values:
[tex]y = 233 (1.06) ^ 2\\y = 262 million[/tex]
Part B:
For the year in which the company reaches 278 million we have:
[tex]278 = 233 (1.06) ^ x[/tex]
Clearing x we have:
[tex](1.06) ^ x = \frac {278} {233}\\(1.06) ^ x = 1.1931\\x = log1.06 (1.1931)\\x = 3[/tex]
Thus, the company reaches 278 million in the year 2009.
Answer:
The equation that models the problem is:
[tex]y = 233 (1.06) ^ x[/tex]
For the year 2008 we have:
[tex]y = 262 million[/tex]
the company reaches 278 million in the year 2009
Using an exponential function, it is found that:
a)
The model is: [tex]y(t) = 233(1.06)^t[/tex].
The estimate of the number of cell phone subscribers in 2008 is of 262 million.
b)
The year when the number of cell phone subscribers was about 278 million was 2009.
What is an exponential function?
An increasing exponential function is modeled by:
[tex]A(t) = A(0)(1 + r)^t[/tex]
In which:
- A(0) is the initial value.
- r is the growth rate, as a decimal.
In this problem:
- In 2006, there was approximately 233 million cell phone subscribers in the United States, hence A(0) = 233.
- During the next 4 years, the number of cell phone subscribers increased by 6% each year, hence r = 0.06.
Then:
[tex]y(t) = A(0)(1 + r)^t[/tex]
[tex]y(t) = 233(1 + 0.06)^t[/tex]
[tex]y(t) = 233(1.06)^t[/tex]
Item a:
2008 is 2 years after 2006, hence:
[tex]y(2) = 233(1.06)^2 = 262[/tex]
The estimate of the number of cell phone subscribers in 2008 is of 262 million.
Item b:
[tex]y(t) = 233(1.06)^t[/tex]
[tex]278 = 233(1.06)^t[/tex]
[tex](1.06)^t = \frac{278}{233}[/tex]
[tex]\log{1.06)^t} = \log{\left(\frac{278}{233}\right)}[/tex]
[tex]t\log{1.06} = \log{\left(\frac{278}{233}\right)}[/tex]
[tex]t = \frac{\log{\left(\frac{278}{233}\right)}}{\log{1.06}}[/tex]
[tex]t = 3.03[/tex]
2006 + 3 = 2009.
The year when the number of cell phone subscribers was about 278 million was 2009.
More can be learned about exponential functions at https://brainly.com/question/25537936
