Suppose you have $\$ 10,000$ to invest. Which of the two rates would yield the larger amount in 5 years: $6 \%$ compounded daily or $5.91 \%$ compounded continuously?
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Which of the two rates would yield the larger amount in 5 years?
$6 \%$ compounded daily
$5.91 \%$ compounded continuously

Step 1 ：Given that the principal amount (P) is $10,000, the time (t) is 5 years, the annual interest rate for the first scenario (r1) is 6% or 0.06, and the annual interest rate for the second scenario (r2) is 5.91% or 0.0591.

Step 2 ：For the first scenario, the interest is compounded daily. Therefore, the number of times that interest is compounded per year (n1) is 365. We can calculate the future value of the investment using the formula for compound interest: A = P(1 + r/n)^(nt). Substituting the given values, we get A1 = 10000(1 + 0.06/365)^(365*5) = 13498.255274436126.

Step 3 ：For the second scenario, the interest is compounded continuously. We can calculate the future value of the investment using the formula for continuous compounding: A = Pe^(rt). Substituting the given values, we get A2 = 10000*e^(0.0591*5) = 13437.980897844.

Step 4 ：Comparing the future values of the two investments, we find that the first investment yields a larger amount in 5 years. The final amounts are approximately $13498.26 and $13437.98 respectively.

Step 5 ：\(\boxed{6 \% \text{ compounded daily}}\) is the better investment option.