Let's solve the problems step-by-step:

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1. Find the interest rate when compounded semiannually

The formula for compound interest is:
$A = P \left( 1 + \frac{r}{n} \right)^{n \cdot t}$

Where:

• $A = 68,211.38$ (future value)
• $P = 55,490$ (initial principal)
• $t = 36 \, \text{months} = 3 \, \text{years}$
• $n = 2$ (compounding semiannually per year)
• $r = \text{interest rate (unknown)}$

Substituting values into the formula:
$68,211.38 = 55,490 \left( 1 + \frac{r}{2} \right)^{2 \cdot 3}$

Simplify:
$\frac{68,211.38}{55,490} = \left( 1 + \frac{r}{2} \right)^6$

$1.22925 = \left( 1 + \frac{r}{2} \right)^6$

Take the 6th root (or raise to the power of $\frac{1}{6}$):
$1 + \frac{r}{2} = 1.22925^{\frac{1}{6}}$

$1 + \frac{r}{2} = 1.0343$

Solve for $r$:
$\frac{r}{2} = 1.0343 - 1 = 0.0343$

$r = 0.0343 \times 2 = 0.0686 \, \text{or} \, 6.86\%$

The interest rate is approximately 6.86% per year.

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2. Find the time required when compounded quarterly

The formula remains:
$A = P \left( 1 + \frac{r}{n} \right)^{n \cdot t}$

Where:

• $A = 50,692.17$
• $P = 45,000$
• r = 0.06 \, \text{(6% annual interest rate)}
• $n = 4 \, \text{(quarterly)}$
• $t = \text{time in years (unknown)}$

Substitute the known values:
$50,692.17 = 45,000 \left( 1 + \frac{0.06}{4} \right)^{4 \cdot t}$

Simplify:
$\frac{50,692.17}{45,000} = \left( 1 + 0.015 \right)^{4 \cdot t}$

$1.12649 = (1.015)^{4 \cdot t}$

Take the natural logarithm on both sides:
$\ln(1.12649) = \ln((1.015)^{4 \cdot t})$

Apply the logarithmic rule $\ln(a^b) = b \ln(a)$:
$\ln(1.12649) = (4 \cdot t) \ln(1.015)$

$t = \frac{\ln(1.12649)}{4 \cdot \ln(1.015)}$

Solve:
$t = \frac{0.1191}{4 \cdot 0.0149} = \frac{0.1191}{0.0596} = 2$

It will take 2 years for the amount to grow from $45,000 to $50,692.17.

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Do you want further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What happens to the time if the interest rate increases in the second problem?
2. How does the compounding frequency affect the interest rate in the first problem?
3. How is semiannual compounding different from quarterly compounding in terms of formula adjustments?
4. How would the answers change if $t$ was given in months instead of years?
5. Can you derive a similar formula for simple interest instead of compound interest?

Tip: Always double-check units (years, months, etc.) to ensure the formula is applied correctly!