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The problem involves compound interest, and we can use the compound interest formula to solve it:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ is the final amount ($2786),
- $P$ is the initial principal ($2000),
- $r$ is the annual interest rate (4.6% = 0.046),
- $n$ is the number of times the interest is compounded per year (quarterly = 4),
- $t$ is the time in years (what we need to find).

Now, we substitute the values and solve for $t$:

\[
2786 = 2000 \left(1 + \frac{0.046}{4}\right)^{4t}
\]

This simplifies to:

\[
2786 = 2000 \left(1 + 0.0115\right)^{4t}
\]

\[
2786 = 2000 (1.0115)^{4t}
\]

Now, divide both sides by 2000:

\[
1.393 = (1.0115)^{4t}
\]

Take the natural logarithm (ln) of both sides:

\[
\ln(1.393) = 4t \cdot \ln(1.0115)
\]

Now, solve for $t$:

\[
t = \frac{\ln(1.393)}{4 \cdot \ln(1.0115)}
\]

Let me calculate this for you.The time it will take for the account to grow to $2786 is approximately **7.25 years**.

Would you like more details on any part of the process, or do you have further questions?

Here are 5 related questions:
1. How would the time change if the interest were compounded monthly instead of quarterly?
2. What happens to the time if the interest rate increases to 5%?
3. How long would it take if you wanted the account to grow to $3000 instead?
4. What is the effect of compounding frequency on the total interest earned over time?
5. Can you explain how continuous compounding differs from quarterly compounding?

**Tip:** The more frequent the compounding, the faster the money grows, but the difference becomes less significant as frequency increases.

Suppose that $2000 is placed in a savings account at an annual rate of 4.6%, compounded quarterly. Assuming that no withdrawals are made, how long will it take for the account to grow to $2786? Do not round any intermediate computations, and round your answer to the nearest hundredth.

Learn how to calculate the time it takes for $2000 to grow to $2786 in a savings account with an annual interest rate of 4.6%, compounded quarterly. This solution covers compound interest, exponential growth, and logarithmic functions. Suitable for high school students in Grades 10-12.

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