To determine the correct equation to model the number of cell phone subscribers in a city [tex]$t$[/tex] years after the year 2000, given that the number of subscribers increases by 60 percent per year, we should follow these steps:
1. Identify the initial number of subscribers:
In the year 2000, the initial number of cell phone subscribers is 200,000.
2. Consider the annual growth rate:
The subscribers increase by 60 percent each year. This rate, as a decimal, is 0.60.
3. Modeling the exponential growth:
In general, the formula for exponential growth is:
[tex]\[
y = P(1 + r)^t
\][/tex]
where:
- [tex]\( y \)[/tex] is the number of subscribers after [tex]\( t \)[/tex] years,
- [tex]\( P \)[/tex] is the initial amount (200,000 subscribers),
- [tex]\( r \)[/tex] is the growth rate per period (0.60),
- [tex]\( t \)[/tex] is the number of time periods (years).
4. Incorporate the given values into the equation:
- Replace [tex]\( P \)[/tex] with 200,000,
- Replace [tex]\( r \)[/tex] with 0.60.
Thus, the correct equation to use is:
[tex]\[
y = 200,000(1 + 0.60)^t
\][/tex]
Let's match this with the given options:
A. [tex]\( y = 200,000(1 + 60)^t \)[/tex] is incorrect because "60" is used instead of "0.60".
B. [tex]\( y = 200,000(1 + 0.6)^t \)[/tex] is correct, as it correctly incorporates a 60 percent increase as a decimal (0.6).
C. [tex]\( y = 200,000(1 - 60)^t \)[/tex] is incorrect because it incorrectly models a decrease (subtraction) and uses "60" instead of "0.60".
D. [tex]\( y = 200,000(1 - 0.6)^t \)[/tex] is incorrect because it represents a decrease by 60 percent rather than an increase.
Therefore, the correct answer is:
[tex]\[
\boxed{y = 200,000(1 + 0.6)^t}
\][/tex]
Hence, the correct choice is:
B. [tex]\( y = 200,000(1 + 0.6)^t \)[/tex]
