How much money should be deposited today in an account that earns $3 \%$ compounded semiannually so that it will accumulate to $\$ 11,000$ in three years?
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The amount of money that should be deposited is $\$$ (Round up to the nearest cent.)
Answer
Round up to the nearest cent to get the final answer: \(\boxed{10059.96}\)
Step 1 ：We are given the future value (FV), the interest rate (r), the number of compounding periods per year (n), and the number of years (t). We need to find the present value (PV). The formula for the present value is: \(PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}\)
Step 2 ：Substitute the given values into the formula: \(FV = 11000\), \(r = 0.03\), \(n = 2\), \(t = 3\)
Step 3 ：Calculate the present value: \(PV = \frac{11000}{(1 + \frac{0.03}{2})^{2*3}}\)
Step 4 ：Simplify the calculation to get the final answer: \(PV = 10059.96411769659\)
Step 5 ：Round up to the nearest cent to get the final answer: \(\boxed{10059.96}\)
