Suppose that $$ \$ 13,000 $$ is invested in a bond fund and the account grows to $$ \$ 14,746.44 $$ in 4 yr . Part: $$ 0 / 2 $$ Part 1 of 2 (a) Use the model $$ A=P e^{\prime \prime} $$ to determine the average rate of return under continuous compounding. Round to the nearest tenth of a percent. Avoid rounding in intermediate steps. \begin{tabular}{ll} The average rate of return under continuous compounding is approximately $$ 3.1 \% $$. \ \hline \end{tabular} Part: 1 / 2 Part 2 of 2 (b) How long will it take the investment to reach $$ \$ 20,000 $$ if the rate of return continues? Round to the nearest tenth of a year. Round values in intermediate steps to three decimal places. It will take the investment approximately $$ \square $$ yr to reach $$ \$ 20,000 $$ if the rate of return continues.

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**Step 1. Write down the formula for continuous compounding.**

We use the formula  
$$
A = P e^{rt}
$$  
where  
$$ A $$ is the amount of money accumulated after time $$ t $$,  
$$ P $$ is the principal (initial investment),  
$$ r $$ is the continuous rate of return, and  
$$ t $$ is time in years.

---

**Step 2. Find the continuous rate $$ r $$.**

We are given that  
$$
14746.44 = 13000 \, e^{4r}.
$$
Divide both sides by $$ 13000 $$:
$$
e^{4r} = \frac{14746.44}{13000} \approx 1.134342.
$$
Take the natural logarithm of both sides:
$$
4r = \ln(1.134342).
$$
Thus,
$$
r = \frac{\ln(1.134342)}{4} \approx \frac{0.12590}{4} \approx 0.031475.
$$
Rounded to the nearest tenth of a percent, the rate $$ r $$ is approximately $$ 3.1\% $$.

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**Step 3. Determine the time $$ t $$ to grow the investment to $$ \$20,000 $$.**

We now set up the equation with $$ P = 13000 $$, $$ A = 20000 $$, and $$ r \approx 0.031475 $$:
$$
20000 = 13000 \, e^{0.031475 \, t}.
$$
Divide both sides by $$ 13000 $$:
$$
e^{0.031475 \, t} = \frac{20000}{13000} \approx 1.538462.
$$
Take the natural logarithm of both sides:
$$
0.031475 \, t = \ln(1.538462).
$$
Calculate the logarithm:
$$
\ln(1.538462) \approx 0.431782.
$$
Now solve for $$ t $$:
$$
t = \frac{0.431782}{0.031475} \approx 13.708.
$$
Rounded to the nearest tenth of a year, $$ t $$ is approximately $$ 13.7 $$ years.

---

$$
\boxed{13.7 \text{ yr}}
$$
It will take approximately 13.7 years for the investment to reach \$20,000 at a continuous compounding rate of 3.1%.

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