Question 11 (1 point) Sara invests $$ \$ 10,000 $$. How long will it take for the investment to double if the rate is $$ 10 \% $$ compounded annually? (Round to the nearest integer) 7 years 8 years 10 months 10 years 7 months

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To determine how long it will take for Sara's investment to double at a rate of $$ 10\% $$ compounded annually, we can use the Rule of 72 or the formula for compound interest.

Using the Rule of 72:
$$
\text{Years to double} \approx \frac{72}{\text{interest rate}}
$$
$$
\text{Years to double} \approx \frac{72}{10} = 7.2 \text{ years}
$$

Now, let's confirm this using the compound interest formula:
$$
A = P(1 + r)^t
$$
where:
- $$ A $$ is the amount of money accumulated after n 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 $$ A = 2P $$:
$$
2P = P(1 + 0.10)^t
$$
Dividing both sides by $$ P $$:
$$
2 = (1.10)^t
$$

Now, we take the logarithm of both sides:
$$
\log(2) = t \cdot \log(1.10)
$$
Solving for $$ t $$:
$$
t = \frac{\log(2)}{\log(1.10)}
$$

Calculating the values:
$$
\log(2) \approx 0.3010
$$
$$
\log(1.10) \approx 0.0414
$$
Now, substituting these values:
$$
t \approx \frac{0.3010}{0.0414} \approx 7.27
$$

Rounding to the nearest integer, it will take approximately **7 years** for the investment to double.

Thus, the answer is:
**7 years**.
It will take approximately 7 years for the investment to double.

Question 11 (1 point) Sara invests \( \$ 10,000 \). How long will it take for the investment to double if the rate is \( 10 \% \) compounded annually? (Round to the nearest integer) 7 years 8 years 10 months 10 years 7 months

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