If \( \$ 37,500 \) is invested at \( 6.4 \% \) for 30 years, find the future value if the interest is compounded the following ways. (Round your answers to the nearest cent.) (a) annually \( \$ \) (b) semiannually \( \$ \) (c) quarterly \( \$ \) (d) monthly \( \$ \) (e) daily ( \( N=360 \) ) \( \$ \) (f) every minute ( \( N=525,600 \) ) \( \$ \) (g) continuously \( \$ \) (h) simple (not compounded)

To find the future value of an investment using different compounding methods, we can use the formulas based on the compounding frequency. Here are the calculations for each scenario given the principal amount \( P = 37,500 \), annual interest rate \( r = 6.4\% = 0.064 \), and time \( t = 30 \) years. ### (a) Annually The formula for future value (FV) with annual compounding is: \[ FV = P(1 + r)^t \] Calculating this gives: \[ FV = 37500(1 + 0.064)^{30} = 37500(1.064)^{30} \approx 37500 \times 6.498 = 243,675.00 \] ### (b) Semiannually The formula for future value with semiannual compounding is: \[ FV = P\left(1 + \frac{r}{n}\right)^{nt} \] where \( n = 2 \): \[ FV = 37500\left(1 + \frac{0.064}{2}\right)^{2 \times 30} = 37500(1.032)^{60} \approx 37500 \times 6.745 = 252,788.75 \] ### (c) Quarterly For quarterly compounding (\( n = 4 \)): \[ FV = P\left(1 + \frac{r}{n}\right)^{nt} \] \[ FV = 37500\left(1 + \frac{0.064}{4}\right)^{4 \times 30} = 37500(1.016)^{120} \approx 37500 \times 6.948 = 260,550.00 \] ### (d) Monthly For monthly compounding (\( n = 12 \)): \[ FV = P\left(1 + \frac{r}{n}\right)^{nt} \] \[ FV = 37500\left(1 + \frac{0.064}{12}\right)^{12 \times 30} = 37500(1.0053333)^{360} \approx 37500 \times 7.067 = 265,753.25 \] ### (e) Daily (N=360) For daily compounding (\( n = 360 \)): \[ FV = P\left(1 + \frac{r}{n}\right)^{nt} \] \[ FV = 37500\left(1 + \frac{0.064}{360}\right)^{360 \times 30} = 37500(1.00017778)^{10800} \approx 37500 \times 7.097 = 266,411.25 \] ### (f) Every minute (N=525,600) For compounding every minute: \[ FV = P\left(1 + \frac{r}{n}\right)^{nt} \] where \( n = 525600 \): \[ FV = 37500\left(1 + \frac{0.064}{525600}\right)^{525600 \times 30} \approx 37500(1.0000001218)^{15768000} \approx 37500 \times 7.105 = 266,633.25 \] ### (g) Continuously For continuous compounding, the formula is: \[ FV = Pe^{rt} \] Now calculating this: \[ FV = 37500e^{0.064 \times 30} \approx 37500 \times e^{1.92} \approx 37500 \times 6.862 = 258,225.75 \] ### (h) Simple (not compounded) For simple interest: \[ FV = P(1 + rt) \] Calculating this gives: \[ FV = 37500(1 + 0.064 \times 30) = 37500(1 + 1.92) = 37500 \times 2.92 = 109,500.00 \] In summary: - (a) \( \$243,675.00 \) - (b) \( \$252,788.75 \) - (c) \( \$260,550.00 \) - (d) \( \$265,753.25 \) - (e) \( \$266,411.25 \) - (f) \( \$266,633.25 \) - (g) \( \$258,225.75 \) - (h) \( \$109,500.00 \)