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Occasionally a savings account may actually pay interest compounded continuously. For each deposit, find the interest earned if interest is compounded (a) semiannually, (b) quarterly, (c) monthly, (d) daily, and (e) continuously. Use 1 year $=365$ days.
Principal
$\$ 1024$
Rate
$1.5 \%$
Time
2 years
(a) The interest earned if interest is compounded semiannually is $\$ 31.07$
(Do not round until the final answer. Then round to the nearest cent as needed.)
(b) The interest earned if interest is compounded quarterly is $\$ 31.13$.
(Do not round until the final answer. Then round to the nearest cent as needed.)
(c) The interest earned if interest is compounded monthly is $\$ \square$.
(Do not round until the final answer. Then round to the nearest cent as needed.)
Answer
\(\boxed{Interest = $31.03}\)
Step 1 ：Given: Principal amount (P) = $1024, annual interest rate (r) = 1.5% = 0.015 (in decimal), time (t) = 2 years, and number of times that interest is compounded per year (n) = 12.
Step 2 ：We use the formula for compound interest: \(A = P(1 + r/n)^{nt}\)
Step 3 ：Substitute the given values into the formula: \(A = 1024(1 + 0.015/12)^{12*2}\)
Step 4 ：Calculate the value inside the parentheses: \(1 + 0.015/12 = 1.00125\)
Step 5 ：Substitute this back into the formula: \(A = 1024(1.00125)^{24}\)
Step 6 ：Calculate the exponent: \((1.00125)^{24} ≈ 1.030454897\)
Step 7 ：Substitute this back into the formula: \(A = 1024 * 1.030454897 ≈ $1055.03\)
Step 8 ：The interest earned is the final amount minus the principal: \(Interest = A - P = $1055.03 - $1024\)
Step 9 ：\(\boxed{Interest = $31.03}\)
