1. **State the problem:** Brayden invests 400 at 9.25% interest compounded continuously. William invests 400 at 8.875% interest compounded monthly. We want to find how much Brayden has when William's money triples. 2. **Formulas:** - Continuous compounding: $$A = P e^{rt}$$ - Monthly compounding: $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where $n=12$ for monthly. 3. **Find time $t$ when William's money triples:** $$3 \times 400 = 400 \left(1 + \frac{0.08875}{12}\right)^{12t}$$ Simplify: $$3 = \left(1 + \frac{0.08875}{12}\right)^{12t}$$ Take natural log: $$\ln(3) = 12t \ln\left(1 + \frac{0.08875}{12}\right)$$ Solve for $t$: $$t = \frac{\ln(3)}{12 \ln\left(1 + \frac{0.08875}{12}\right)}$$ Calculate: $$1 + \frac{0.08875}{12} = 1 + 0.0073958333 = 1.0073958333$$ $$\ln(3) \approx 1.098612$$ $$\ln(1.0073958333) \approx 0.007368$$ $$t = \frac{1.098612}{12 \times 0.007368} = \frac{1.098612}{0.088416} \approx 12.42 \text{ years}$$ 4. **Calculate Brayden's amount at $t=12.42$ years:** $$A = 400 e^{0.0925 \times 12.42}$$ Calculate exponent: $$0.0925 \times 12.42 = 1.14885$$ Calculate: $$A = 400 e^{1.14885}$$ $$e^{1.14885} \approx 3.154$$ $$A = 400 \times 3.154 = 1261.6$$ 5. **Final answer:** Brayden will have approximately **1262** dollars when William's money has tripled.