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? 20 years (Round to the nearest whole number.) d. How long will Jamal have to wait to become a millionaire if he invests $$ \$ 284,998.58 $$ at $$ 7 \% $$ today? $$ \square $$ years (Round to the nearest whole number.)

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To determine how long Jamal needs to wait to become a millionaire by investing **\$284,998.58** at an annual interest rate of **7%**, we can use the **compound interest formula**:

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

Where:
- $$ A $$ = the amount of money accumulated after $$ t $$ years (in this case, \$1,000,000)
- $$ P $$ = the principal investment amount (\$284,998.58)
- $$ r $$ = annual interest rate (7% or 0.07)
- $$ t $$ = the number of years

**Step-by-Step Calculation:**

1. **Set up the equation:**

$$
1,000,000 = 284,998.58 \times (1 + 0.07)^t
$$

2. **Divide both sides by \$284,998.58 to isolate the growth factor:**

$$
\frac{1,000,000}{284,998.58} = (1.07)^t
$$

$$
3.509 = (1.07)^t
$$

3. **Take the natural logarithm of both sides to solve for $$ t $$:**

$$
\ln(3.509) = t \times \ln(1.07)
$$

4. **Calculate the logarithms:**

$$
\ln(3.509) \approx 1.25276
$$
$$
\ln(1.07) \approx 0.067658
$$

5. **Solve for $$ t $$:**

$$
t = \frac{1.25276}{0.067658} \approx 18.5 \text{ years}
$$

6. **Round to the nearest whole number:**

$$
t \approx 19 \text{ years}
$$

**Conclusion:**

Jamal will need to wait approximately **19 years** to become a millionaire with an investment of \$284,998.58 at an annual interest rate of 7%.

$$
\boxed{19}
$$
Jamal will need to wait approximately 19 years to become a millionaire.

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