Let's break down the problem step by step.

Given Information:

The equation for the amount $A$ in the account is related to the compound interest formula, where:

• $P$ is the initial principal,
• $r$ is the annual interest rate,
• $n$ is the number of compounding periods per year,
• $t$ is the time in years.

The general form of the compound interest formula is:

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

Option a: $A = 2000\left(1 + \frac{0.07125}{12}\right)^{12t}$

• Initial Principal ($P$) = 2000
• Annual Interest Rate ($r$) = 0.07125 (or 7.125%)
• Number of Compounding Periods per Year ($n$) = 12 (monthly compounding)

This equation represents compound interest with monthly compounding. The formula is:

$A = 2000 \left( 1 + \frac{0.07125}{12} \right)^{12t}$

a) Amount in the account after 5 years:

To find the amount $A$ after 5 years, substitute $t = 5$ into the equation:

$A = 2000 \left( 1 + \frac{0.07125}{12} \right)^{12 \times 5}$

Let’s calculate it:

$A = 2000 \left( 1 + 0.0059375 \right)^{60}$

$A = 2000 \left( 1.0059375 \right)^{60}$

$A \approx 2000 \times 1.42576$

$A \approx 2851.52$

So, after 5 years, the amount in the account will be approximately $2851.52.

b) Time required for the investment to double:

To find how long it takes for the initial investment to double, we set $A = 2P$, meaning the amount in the account is double the initial investment. So, the equation becomes:

$4000 = 2000 \left( 1 + \frac{0.07125}{12} \right)^{12t}$

Simplify:

$2 = \left( 1 + \frac{0.07125}{12} \right)^{12t}$

$2 = \left( 1.0059375 \right)^{12t}$

Now, take the natural logarithm (ln) of both sides:

$\ln(2) = 12t \times \ln(1.0059375)$

$0.6931 = 12t \times 0.005933$

Solve for $t$:

$t = \frac{0.6931}{12 \times 0.005933}$

$t \approx \frac{0.6931}{0.071196}$

$t \approx 9.73 \text{ years}$

So, it will take approximately 9.73 years for the initial investment to double.

---

Recap of the Answers:

• After 5 years, the amount in the account is approximately $2851.52.
• Time to double the investment is approximately 9.73 years.

---

Do you need any further clarification or additional details?

Additional Questions:

1. What is the impact of increasing the interest rate on the time to double the investment?
2. How would the amount in the account change if compounding were done quarterly instead of monthly?
3. If the initial investment were $2500, how would this affect the final amount after 5 years?
4. How can the formula be adjusted if the interest is compounded continuously instead of periodically?
5. How would the time to double the investment change if the annual interest rate were 10% instead of 7.125%?

Tip:

The more frequently the interest is compounded, the more interest accumulates over time due to the effects of compounding.