In its first 10 years a mulual fund produced an average annual return of $$ 19.15 \% $$. Assume that money invested in this fund continues to earn $$ 19.15 \% $$ compounded annually. How long will it take money invested in this fund to double? It will take approximately $$ \square $$ year(s) for the money invested in this fund to double. (Round up to the nearest year.)

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To determine how long it will take for the money invested in the mutual fund to double at an average annual return of $$ 19.15\% $$, we can use the Rule of 72 or the formula for compound interest. However, since we need a precise calculation, we'll use the compound interest formula:

$$
A = P(1 + r)^t
$$

Where:
- $$ A $$ is the amount of money accumulated after $$ t $$ years, including interest.
- $$ P $$ is the principal amount (the initial amount of money).
- $$ r $$ is the annual interest rate (decimal).
- $$ t $$ is the number of years the money is invested or borrowed.

In this case, we want to find $$ t $$ when $$ A = 2P $$ (the money doubles). Thus, we can set up the equation:

$$
2P = P(1 + 0.1915)^t
$$

We can simplify this equation by dividing both sides by $$ P $$:

$$
2 = (1 + 0.1915)^t
$$

Now, we can rewrite it as:

$$
2 = (1.1915)^t
$$

To solve for $$ t $$, we can take the logarithm of both sides:

$$
\log(2) = t \cdot \log(1.1915)
$$

Now, we can solve for $$ t $$:

$$
t = \frac{\log(2)}{\log(1.1915)}
$$

Let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\log_{10}{\left(2\right)}}{\log_{10}{\left(1.1915\right)}}$$
- step1: Convert the expressions:
  $$\frac{\log_{10}{\left(2\right)}}{\log_{10}{\left(\frac{2383}{2000}\right)}}$$
- step2: Use the logarithm base change rule:
  $$\log_{\frac{2383}{2000}}{\left(2\right)}$$
Calculating the value gives us:

$$
t \approx 3.956
$$

Rounding up to the nearest year, it will take approximately $$ 4 $$ years for the money invested in this fund to double.

Thus, the final answer is:

It will take approximately $$ 4 $$ year(s) for the money invested in this fund to double.
It will take approximately 4 years for the money invested in the mutual fund to double.

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