A certain type of account must be opened with an investment for a positive integer number of years. No other deposits or withdrwals are permitted, and the account earns 7 percent intest that is compounded annually. If $400 were invested in the account for x years, what is the smallest possible value of x such that at the end of x years, the amount in the account will be least 3 times initial investment ?

Options

A. 14
B. 15
C. 16
D. 17

Explanation

To solve the problem, we need to determine the smallest integer value of \( x \) such that the amount in the account after \( x \) years is at least three times the initial investment of $400, given that the account earns 7% interest compounded annually.

### Step-by-Step Solution:

1. **Identify the initial investment and target amount**:
- Initial investment, \( P = 400 \) dollars.
- Target amount, which is at least three times the initial investment:
\[ 3 \times 400 = 1200 \text{ dollars}. \]

2. **Use the compound interest formula**:
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^x \]
where \( A \) is the amount after \( x \) years, \( P \) is the principal, \( r \) is the rate of interest, and \( x \) is the number of years.

3. **Substitute the known values into the formula**:
We need to find \( x \) such that:
\[ 400 \left(1 + \frac{7}{100}\right)^x \geq 1200. \]
Simplifying this gives:
\[ 400 \left(1.07\right)^x \geq 1200. \]

4. **Divide both sides by 400**:
\[ \left(1.07\right)^x \geq \frac{1200}{400}. \]
This simplifies to:
\[ \left(1.07\right)^x \geq 3. \]

5. **Take the logarithm of both sides**:
\[ \log\left((1.07)^x\right) \geq \log(3). \]
Using the property of logarithms, this can be rewritten as:
\[ x \cdot \log(1.07) \geq \log(3). \]

6. **Solve for \( x \)**:
\[ x \geq \frac{\log(3)}{\log(1.07)}. \]
Now we need to calculate the values of \( \log(3) \) and \( \log(1.07) \):
- \( \log(3) \approx 0.4771 \)
- \( \log(1.07) \approx 0.0291 \)

Plugging these values into the inequality:
\[ x \geq \frac{0.4771}{0.0291} \approx 16.38. \]

7. **Determine the smallest integer value of \( x \)**:
Since \( x \) must be a positive integer, we round up \( 16.38 \) to the next whole number:
\[ x = 17. \]

### Conclusion:
The smallest possible value of \( x \) such that the amount in the account will be at least three times the initial investment is \( \boxed{17} \).