suppose you want to have $800,000 for retirement in 30 years. your account earns 8% interest.\na) how much…

Explanation:

Step1: Identify the relevant formula for monthly - deposit in a future - value of an ordinary annuity

The formula for the future value of an ordinary annuity is $F = A\times\frac{(1 + r)^{n}-1}{r}$, where $F$ is the future value of the annuity, $A$ is the amount of each payment (monthly deposit), $r$ is the interest rate per period, and $n$ is the number of periods. The annual interest rate $i = 8%=0.08$. The monthly interest rate $r=\frac{0.08}{12}$, and the number of years $t = 30$. The number of periods $n=30\times12 = 360$ months, and $F=$800000$. We need to solve the formula for $A$: $A=\frac{F\times r}{(1 + r)^{n}-1}$.

Step2: Calculate the monthly deposit $A$

Substitute the values into the formula: $r=\frac{0.08}{12}\approx0.00667$, $n = 360$, $F = 800000$. $A=\frac{800000\times\frac{0.08}{12}}{(1+\frac{0.08}{12})^{360}-1}$ First, calculate $(1+\frac{0.08}{12})^{360}$. Let $x=\frac{0.08}{12}\approx0.00667$, then $(1 + x)^{360}=(1.00667)^{360}\approx10.9357$. $A=\frac{800000\times0.00667}{10.9357 - 1}=\frac{5336}{9.9357}\approx537.05$.

Step3: Calculate the total amount deposited

The total amount deposited over 30 years (360 months) is $T = A\times n=537.05\times360 = 193338$.

Step4: Calculate the interest earned

The interest earned $I$ is the future value minus the total amount deposited. $I=F - T=800000-193338 = 606662$.

Answer:

a) $537.05 b) $606662