Question

a bank offers the following two investment options. find the value for each investment option if $20,000 is invested for 8 years. assume the full amount is withdrawn. long - term investment! 10 - year cd at 2.785% apy! enjoy online or at one of our convenient locations! note: cd means certificate of deposit. apy=(1 + \frac{r}{12})^{12}-1. early withdrawal fee before 10 years is 2% of account balance. money maker savings! minimum balance: $10,000. earn 2.5% interest compounded monthly. loyalty program! every 4 years with us, your interest rate increases by 0.25%! the value of the long - term investment is $ and the value of the money maker savings is $ (round to the nearest dollar as needed.)

Explanation:

Step1: Calculate the value of the Long - Term Investment (CD)

The formula for compound - interest is $A = P(1 + r)^{n}$, where $P$ is the principal amount, $r$ is the interest rate per period, and $n$ is the number of periods. For the 10 - year CD with an APY (Annual Percentage Yield) of $2.785\%=0.02785$, and $P = 20000$, $n = 10$.
$A_{CD}=20000\times(1 + 0.02785)^{10}$
$A_{CD}=20000\times1.02785^{10}$
Using a calculator, $1.02785^{10}\approx1.31177$.
$A_{CD}=20000\times1.31177 = 26235.4\approx26235$

Step2: Calculate the value of the Money Maker Savings

The initial interest rate $r_0=2.5\% = 0.025$, compounded monthly ($m = 12$). The number of years $t = 8$.
The compound - interest formula is $A=P(1+\frac{r}{m})^{mt}$.
For the first 4 years:
$A_1 = 20000(1+\frac{0.025}{12})^{12\times4}$
$A_1=20000(1+\frac{0.025}{12})^{48}$
Let $x=\frac{0.025}{12}\approx0.0020833$, then $(1 + x)^{48}\approx1.105167$.
$A_1=20000\times1.105167 = 22103.34$

For the next 4 years, the interest rate increases by $0.25\%$, so $r_1=2.5\%+0.25\% = 2.75\%=0.0275$
$A_2=A_1(1+\frac{0.0275}{12})^{12\times4}$
$A_2 = 22103.34(1+\frac{0.0275}{12})^{48}$
Let $y=\frac{0.0275}{12}\approx0.0022917$, then $(1 + y)^{48}\approx1.14094$.
$A_2=22103.34\times1.14094\approx25217$

Answer:

The value of the Long - Term Investment is $\$26235$ and the value of the Money Maker Savings is $\$25217$