A marketing team conducted a study on the use of smartphones. In a certain metropolitan area, there were 1.6 million smartphone users at the end of 2018. The marketing team predicted that the number of smartphone users would increase by 35 percent each year beginning in 2019. If y represents the number of smartphone users in this metropolitan area after x years, then which of the following equations best models the number of smartphone users in this area over time ?

Options

A. `y=1,600,000(1.35)^(x)`
B. `y=1,600,000(35)^(x)`
C. `y=35+1,600,000`
D. `y = 1.35x + 1,600, 000`

Answer

To model the number of smartphone users in a metropolitan area over time, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial Users**:
At the end of 2018, the number of smartphone users is given as 1.6 million. This can be represented as:
\[ A_0 = 1.6 \text{ million} = 1,600,000 \]

2. **Determine the Growth Rate**:
The problem states that the number of smartphone users is expected to increase by 35% each year. This percentage can be converted into a decimal for calculations:
\[ R = 35\% = \frac{35}{100} = 0.35 \]

3. **Set Up the Growth Formula**:
The formula for exponential growth when a quantity increases by a certain percentage each year is:
\[ Y = A_0 \times (1 + R)^x \]
where \(Y\) is the number of users after \(x\) years, \(A_0\) is the initial quantity, \(R\) is the growth rate, and \(x\) is the number of years.

4. **Substitute the Known Values**:
Now, substitute \(A_0\) and \(R\) into the formula:
\[ Y = 1,600,000 \times (1 + 0.35)^x \]
This simplifies to:
\[ Y = 1,600,000 \times (1.35)^x \]

5. **Final Equation**:
Thus, the equation that models the number of smartphone users in the metropolitan area after \(x\) years is:
\[ Y = 1,600,000 \times (1.35)^x \]

### Conclusion:
The equation \(Y = 1,600,000 \times (1.35)^x\) best represents the number of smartphone users in this area over time.