To solve this problem, we need to first determine how long it will take for Brianna's investment to triple, and then calculate how much William's investment would grow in that same amount of time.

Step 1: Find how long it takes for Brianna's investment to triple.

Brianna's investment is compounded monthly, so we use the formula for compound interest:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• $A$ is the amount after time $t$,
• $P$ is the principal (initial amount),
• $r$ is the annual interest rate (as a decimal),
• $n$ is the number of times the interest is compounded per year,
• $t$ is the time in years.

Brianna's values:

• $P = 33,000$,
• $r = 5.5\% = 0.055$,
• $n = 12$ (monthly compounding),
• $A = 3P = 99,000$ (since the money triples).

Substituting these values into the formula:

$99,000 = 33,000 \left(1 + \frac{0.055}{12}\right)^{12t}$

Now, solve for $t$:

$3 = \left(1 + \frac{0.055}{12}\right)^{12t}$

$3 = \left(1 + 0.0045833\right)^{12t}$

$3 = (1.0045833)^{12t}$

Take the natural logarithm of both sides:

$\ln(3) = 12t \ln(1.0045833)$

$\ln(3) \approx 1.0986 \quad \text{and} \quad \ln(1.0045833) \approx 0.004577$

Now solve for $t$:

$1.0986 = 12t \times 0.004577$

$t = \frac{1.0986}{12 \times 0.004577}$

$t \approx \frac{1.0986}{0.054924}$

$t \approx 20.0 \text{ years}$

So, it will take Brianna about 20 years for her investment to triple.

Step 2: Calculate how much William's investment grows in 20 years.

For William's investment, which is compounded continuously, we use the formula:

$A = P e^{rt}$

Where:

• $A$ is the amount after time $t$,
• $P$ is the principal (initial amount),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time in years,
• $e$ is the base of the natural logarithm.

William's values:

• $P = 33,000$,
• $r = 5.125\% = 0.05125$,
• $t = 20$ years.

Substitute these into the formula:

$A = 33,000 e^{0.05125 \times 20}$

$A = 33,000 e^{1.025}$

Using the value of $e^{1.025} \approx 2.783$:

$A = 33,000 \times 2.783$

$A \approx 91,979$

So, to the nearest dollar, William would have $91,979 in his account when Brianna's money has tripled.

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Summary:

• Brianna's money will triple in about 20 years.
• After that time, William's investment will be approximately $91,979.

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Would you like further details on any of the steps? Here are some related questions for exploration:

1. How would the result change if Brianna's interest rate were compounded annually instead of monthly?
2. What would happen to the growth of William's account if the interest rate increased to 6%?
3. How do we calculate compound interest for different compounding frequencies (quarterly, daily)?
4. How does continuous compounding compare to monthly compounding in terms of growth?
5. What happens if we change the initial investment amounts?

Tip: For continuous compounding, the formula $A = P e^{rt}$ provides a very smooth growth curve compared to the stepped growth with periodic compounding.