To determine the number of years, [tex]\( t \)[/tex], it takes for Tim to earn \[tex]$200 in the given bank account, we follow these steps:
1. Identify the known variables:
- \( P \): Initial deposit, which is \$[/tex]200.
- [tex]\( r \)[/tex]: Annual interest rate, which is [tex]\(0.01\)[/tex] (or [tex]\(1\%\)[/tex]).
- [tex]\( n \)[/tex]: Number of times the interest is compounded per year, which is monthly, so [tex]\(n = 12\)[/tex].
- [tex]\( A \)[/tex]: Desired amount, which is \[tex]$200.
2. Write the formula for compound interest:
The formula for compound interest is:
\[
S(t) = P \left(1 + \frac{r}{n}\right)^{nt}
\]
3. Set up the equation with the known amounts:
We need to solve for \( t \) when the final amount \( S(t) \) is \$[/tex]200:
[tex]\[
200 = 200 \left(1 + \frac{0.01}{12}\right)^{12t}
\][/tex]
4. Simplify the equation:
Dividing both sides by 200, we get:
[tex]\[
1 = \left(1 + \frac{0.01}{12}\right)^{12t}
\][/tex]
Simplifying further:
[tex]\[
1 = \left(1 + \frac{1}{1200}\right)^{12t}
\][/tex]
Note that [tex]\(1 + \frac{1}{1200} \approx 1.000833\)[/tex].
5. Solve for [tex]\( t \)[/tex]:
Since the base of the exponent on both sides of the equation is 1, we can conclude that the exponential part [tex]\( (1.000833)^{12t} \)[/tex] must be equal to 1. For any positive base different from 1, the exponent must be 0 for the result to be 1. Therefore:
[tex]\[
12t = 0
\][/tex]
Solving for [tex]\( t \)[/tex], we have:
[tex]\[
t = 0
\][/tex]
6. Determine if Tim should switch banks:
We found that it takes [tex]\( t = 0 \)[/tex] years for Tim to earn \[tex]$200 at this bank account with the given conditions. This implies there's no additional time needed for his savings to reach \$[/tex]200 from his initial deposit of \[tex]$200. Hence, the time required is less than any desired number of years, and Tim should switch to this bank account, as evidenced by the stated conditions.
So, the detailed, step-by-step solution is:
- It will take Tim \( 0 \) years to earn \$[/tex]200 at this bank (essentially, he already has the desired amount without any waiting period).
- Tim should switch to this bank with the compounded monthly interest rate.
