For the following amount at the given interest rate compounded continuously, find (a) the future value after 7 years, (b) the effective rate, and (c) the time to reach $\$ 11,000$.
$\$ 5200$ at $3.3 \%$
a. The future value after 7 years is approximately $\$ 6551.27$.
(Do not round until the final answer. Then round to the nearest cent as needed.)
b. The effective rate is $3.36 \%$.
(Do not round until the final answer. Then round to two decimal places as needed.)
c. The time to reach $\$ 11,000$ from $\$ 5200$ at $3.3 \%$ is $\square$ years.
(Do not round until the final answer. Then round to two decimal places as needed.)
Answer
So, the final answers are: (a) The future value after 7 years is approximately \(\boxed{6551.27}\) dollars, (b) The effective rate is approximately \(\boxed{3.36}\)% , and (c) The time to reach $11,000 from $5200 at 3.3% is approximately \(\boxed{22.70}\) years.
Step 1 ：We are given a principal amount of $5200 and an interest rate of 3.3% compounded continuously. We are asked to find (a) the future value after 7 years, (b) the effective rate, and (c) the time to reach $11,000.
Step 2 ：First, let's find the future value after 7 years. The formula for future value with continuous compounding is \(A = P e^{rt}\), where \(P\) is the principal amount, \(r\) is the interest rate, \(t\) is the time in years, and \(e\) is the base of the natural logarithm.
Step 3 ：Substituting the given values into the formula, we get \(A = 5200 e^{0.033 \times 7}\).
Step 4 ：Calculating the above expression, we find that the future value after 7 years is approximately $6551.27.
Step 5 ：Next, let's find the effective rate. The formula for the effective rate with continuous compounding is \(r_{eff} = e^r - 1\), where \(r\) is the nominal interest rate and \(e\) is the base of the natural logarithm.
Step 6 ：Substituting the given interest rate into the formula, we get \(r_{eff} = e^{0.033} - 1\).
Step 7 ：Calculating the above expression, we find that the effective rate is approximately 3.36%.
Step 8 ：Finally, let's find the time to reach $11,000. Rearranging the formula for future value, we get \(t = \frac{\ln(A/P)}{r}\), where \(\ln\) is the natural logarithm.
Step 9 ：Substituting the given values into the formula, we get \(t = \frac{\ln(11000/5200)}{0.033}\).
Step 10 ：Calculating the above expression, we find that the time to reach $11,000 from $5200 at 3.3% is approximately 22.70 years.
Step 11 ：So, the final answers are: (a) The future value after 7 years is approximately \(\boxed{6551.27}\) dollars, (b) The effective rate is approximately \(\boxed{3.36}\)% , and (c) The time to reach $11,000 from $5200 at 3.3% is approximately \(\boxed{22.70}\) years.
