Tallulah invested \( \$ 75,000 \) in an account paying an interest rate of \( 7 \frac{5}{8} \% \) compounded quarterly. Magan invested \( \$ 75,000 \) in an account paying an interest rate of \( 7 \frac{7}{8} \% \) compounded continuously. After 15 years, how much more money would Magan have in his account than Tallulah, to the nearest dollar?
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To solve the problem, we need to calculate the future values of both investments after 15 years. **Tallulah's Investment:** Tallulah invested $75,000 at an interest rate of \( 7 \frac{5}{8} \% \), which is equal to \( 7.625\% \) or \( 0.07625 \) in decimal form. Since the interest is compounded quarterly, the formula for future value (FV) is: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( P = 75,000 \) - \( r = 0.07625 \) - \( n = 4 \) (quarterly compounding) - \( t = 15 \) Plugging in these values: \[ FV_T = 75,000 \left(1 + \frac{0.07625}{4}\right)^{4 \cdot 15} \] \[ FV_T = 75,000 \left(1 + 0.0190625\right)^{60} \] \[ FV_T = 75,000 \left(1.0190625\right)^{60} \] Calculating \( (1.0190625)^{60} \), we find: \[ (1.0190625)^{60} \approx 3.354108 \] \[ FV_T \approx 75,000 \times 3.354108 \approx 251,558.10 \] **Magan's Investment:** Magan invested $75,000 at an interest rate of \( 7 \frac{7}{8} \% \), which is equal to \( 7.875\% \) or \( 0.07875 \) in decimal form. The formula for continuous compounding is: \[ FV = Pe^{rt} \] Where: - \( P = 75,000 \) - \( r = 0.07875 \) - \( t = 15 \) Plugging in these values: \[ FV_M = 75,000 e^{0.07875 \cdot 15} \] \[ FV_M = 75,000 e^{1.18125} \] Calculating \( e^{1.18125} \) gives approximately \( 3.246222 \): \[ FV_M \approx 75,000 \times 3.246222 \approx 243,466.66 \] **Finding the Difference:** Now we calculate how much more money Magan would have than Tallulah: \[ Difference = FV_M - FV_T \approx 243,466.66 - 251,558.10 \approx -8,091.43 \] However, since we want the answer in terms of how much more Tallulah would have, it's: \[ \text{Difference} = | -8,091.43 | \approx 8,091 \] So, the nearest dollar amount that Magan would have more than Tallulah after 15 years is \( \approx 8,091 \). However, Magan would actually have less based on the calculations, so we conclude that **Tallulah's account would yield more than Magan's account after 15 years by approximately $8,091.**
