An initial investment amount $P$, an annual interest rate $r$, and a time $t$ are given. Find the future value of the investment when interest is compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. Then find $(e)$ the doubling time $T$ for the given interest rate.
\[
P=\$ 7500, r=2.45 \%, t=6 y r
\]
Answer
Final Answer: \[\boxed{A_{\text{annually}} = \$8687.65, A_{\text{monthly}} = \$8687.65, A_{\text{daily}} = \$8687.65, A_{\text{continuously}} = \$8687.65, T = 28.29 \text{ years}}\]
Step 1 ：Given an initial investment amount $P = \$7500$, an annual interest rate $r = 2.45\% = 0.0245$ (in decimal form), and a time $t = 6$ years, we are asked to find the future value of the investment when interest is compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. Then find (e) the doubling time $T$ for the given interest rate.
Step 2 ：The future value of an investment can be calculated using the formula for compound interest: \[A = P(1 + \frac{r}{n})^{nt}\] where: \[A\] is the amount of money accumulated after n years, including interest, \[P\] is the principal amount (the initial amount of money), \[r\] is the annual interest rate (in decimal form), \[n\] is the number of times that interest is compounded per year, and \[t\] is the time the money is invested for in years.
Step 3 ：For the different compounding frequencies, we will use: Annually: n = 1, Monthly: n = 12, Daily: n = 365.
Step 4 ：For continuous compounding, the formula is slightly different: \[A = Pe^{rt}\] where e is the base of the natural logarithm, approximately equal to 2.71828.
Step 5 ：Finally, the doubling time can be found using the formula: \[T = \frac{\ln(2)}{r}\] where ln is the natural logarithm.
Step 6 ：Calculating these values, we find that the future value of the investment when interest is compounded annually is approximately \$8687.65, when compounded monthly is approximately \$8687.65, when compounded daily is approximately \$8687.65, and when compounded continuously is approximately \$8687.65. The doubling time for the given interest rate is approximately 28.29 years.
Step 7 ：Final Answer: \[\boxed{A_{\text{annually}} = \$8687.65, A_{\text{monthly}} = \$8687.65, A_{\text{daily}} = \$8687.65, A_{\text{continuously}} = \$8687.65, T = 28.29 \text{ years}}\]
