Part (a): Write the function for the balance.

The formula for compound interest is:

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

Where:

• $A(t)$ is the amount of money in the account after $t$ years.
• $P$ is the initial principal (the amount of money you started with), which is $9000.
• $r$ is the annual interest rate (3.6% = 0.036).
• $n$ is the number of times the interest is compounded per year (monthly compounding means $n = 12$).
• $t$ is the time in years.

Plugging the known values into the formula:

$s(t) = 9000 \left(1 + \frac{0.036}{12}\right)^{12t}$

This is the function that represents the balance in the savings account after $t$ years.

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Part (b): Find the balance after 7 years.

To find the balance after 7 years, substitute $t = 7$ into the formula:

$s(7) = 9000 \left(1 + \frac{0.036}{12}\right)^{12 \times 7}$

Now calculate:

$s(7) = 9000 \left(1 + 0.003\right)^{84} = 9000 \left(1.003\right)^{84}$

We need to calculate $1.003^{84}$:

$1.003^{84} \approx 1.2668$

Then:

$s(7) = 9000 \times 1.2668 \approx 11,401.20$

So, the balance after 7 years is approximately $11,401.

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Part (c): Find the time to double the principal.

To double the principal, we want $A(t) = 2 \times P = 2 \times 9000 = 18,000$.

Using the compound interest formula, set $A(t) = 18,000$:

$18000 = 9000 \left(1 + \frac{0.036}{12}\right)^{12t}$

Simplify:

$2 = \left(1.003\right)^{12t}$

Take the natural logarithm (ln) of both sides:

$\ln(2) = \ln\left(1.003^{12t}\right)$

Using the logarithmic property $\ln(a^b) = b \ln(a)$, this becomes:

$\ln(2) = 12t \ln(1.003)$

Now, calculate the values:

$\ln(2) \approx 0.6931 \quad \text{and} \quad \ln(1.003) \approx 0.002999$

So:

$0.6931 = 12t \times 0.002999$

Solve for $t$:

$t = \frac{0.6931}{12 \times 0.002999} \approx \frac{0.6931}{0.035988} \approx 19.3$

Thus, it will take about 19.3 years to double the principal.

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Summary of answers:

• (a) The function for the balance is:
  $s(t) = 9000 \left(1 + \frac{0.036}{12}\right)^{12t}$

• (b) The balance after 7 years is about $11,401.

• (c) It will take about 19.3 years to double the principal.

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Would you like further details on any of these steps or have additional questions?

Here are some related questions:

1. How does changing the compounding frequency affect the balance?
2. What would happen to the time it takes to double the principal if the interest rate was higher?
3. How would you adjust the formula if the interest were compounded quarterly instead of monthly?
4. If you withdrew money at the end of each year, how would the formula change?
5. How does the initial deposit affect the final balance in compound interest scenarios?

Tip: When working with compound interest problems, always keep track of the units (years, months, etc.) to ensure consistency in the formula.