Let's break down the problem step-by-step and address each part:
Part (A): What values should be used for [tex]\( a \)[/tex], [tex]\( r \)[/tex], and [tex]\( k \)[/tex]?
- [tex]\( a \)[/tex] is the principal amount that Manuela deposits, which is $2000.
- [tex]\( r \)[/tex] is the annual percentage rate, which is 3.4%. In decimal form, this is [tex]\( \frac{3.4}{100} = 0.034 \)[/tex].
- [tex]\( k \)[/tex] is the number of times interest is compounded per year, which is quarterly. Thus, [tex]\( k \)[/tex] is 4.
So, the values are:
[tex]\[ a = 2000 \][/tex]
[tex]\[ r = 0.034 \][/tex]
[tex]\[ k = 4 \][/tex]
Part (B): How much money will Manuela have in the account in 10 years?
Using the exponential formula for compound interest:
[tex]\[ A(t) = a \left(1 + \frac{r}{k}\right)^{kt} \][/tex]
We plug in the values:
[tex]\[ a = 2000 \][/tex]
[tex]\[ r = 0.034 \][/tex]
[tex]\[ k = 4 \][/tex]
[tex]\[ t = 10 \][/tex]
Substitute these values into the formula:
[tex]\[ A(10) = 2000 \left(1 + \frac{0.034}{4}\right)^{4 \cdot 10} \][/tex]
Calculate inside the parentheses first:
[tex]\[ 1 + \frac{0.034}{4} = 1.0085 \][/tex]
Now raise this to the power of [tex]\( 4 \cdot 10 = 40 \)[/tex]:
[tex]\[ 1.0085^{40} \approx 1.40293032224716 \][/tex]
Finally, multiply by the principal amount:
[tex]\[ A(10) = 2000 \times 1.40293032224716 = 2805.86064449432 \][/tex]
Rounded to the nearest penny, the amount is:
[tex]\[ \boxed{2805.86} \][/tex]
Part (C): What is the annual percentage yield (APY) for the savings account?
The APY is calculated as follows:
[tex]\[ APY = \left( \left(1 + \frac{r}{k}\right)^k - 1 \right) \times 100 \% \][/tex]
Substitute the values:
[tex]\[ r = 0.034 \][/tex]
[tex]\[ k = 4 \][/tex]
[tex]\[ APY = \left( \left(1 + \frac{0.034}{4}\right)^4 - 1 \right) \times 100 \% \][/tex]
[tex]\[ = \left( \left(1.0085\right)^4 - 1 \right) \times 100 \][/tex]
[tex]\[ = \left(1.034435961720062247 - 1\right) \times 100 \][/tex]
[tex]\[ = 0.034435961720062247 \times 100 \][/tex]
[tex]\[ = 3.4435961720062247 \% \][/tex]
Rounded to three decimal places, the APY is:
[tex]\[ \boxed{3.444 \%} \][/tex]
