Waiting period. Jamal is waiting to be a millionaire. He wants to know how long he must wait if a. he invests $$ \$ 27,523.17 $$ at $$ 20 \% $$ today? b. he invests $$ \$ 54,955.85 $$ at $$ 13 \% $$ today? c. he invests $$ \$ 126,545.56 $$ at $$ 11 \% $$ today? d. he invests $$ \$ 284,998.58 $$ at $$ 7 \% $$ today? 24 years (Round to the nearest whole number.) c. How long will Jamal have to wait to become a millionaire if he invests $$ \$ 126,545.56 $$ at $$ 11 \% $$ today? years (Round to the nearest whole number.)

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To determine how long Jamal must wait to become a millionaire by investing \$126,545.56 at an annual interest rate of 11%, we'll use the compound interest formula:

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

Where:
- $$ A $$ is the amount of money accumulated after $$ t $$ years, including interest.
- $$ P $$ is the principal investment amount (\$126,545.56).
- $$ r $$ is the annual interest rate (11% or 0.11).
- $$ t $$ is the number of years.

We want to find $$ t $$ when $$ A = \$1,000,000 $$:

$$
1,000,000 = 126,545.56 \times (1 + 0.11)^t
$$

Divide both sides by \$126,545.56:

$$
\frac{1,000,000}{126,545.56} = (1.11)^t
$$

$$
7.9007 \approx (1.11)^t
$$

To solve for $$ t $$, take the natural logarithm (ln) of both sides:

$$
\ln(7.9007) = t \times \ln(1.11)
$$

$$
2.066 = t \times 0.10436
$$

$$
t = \frac{2.066}{0.10436} \approx 19.78
$$

Rounding to the nearest whole number:

$$
t \approx 20 \text{ years}
$$

**Answer:** Jamal will need to wait **20 years** to become a millionaire with this investment.
Jamal will need to wait **20 years** to become a millionaire by investing \$126,545.56 at an 11% annual interest rate.

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