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A bank features a savings account that has an annual percentage rate of [tex]\( r = 3.4\% \)[/tex] with interest compounded quarterly. Manuela deposits [tex]\( \$2,000 \)[/tex] into the account.

The account balance can be modeled by the exponential formula
[tex]\[ A(t) = a \left(1 + \frac{r}{k} \right)^{kt} \][/tex]
where [tex]\( A \)[/tex] is the account value after [tex]\( t \)[/tex] years, [tex]\( a \)[/tex] is the principal (starting amount), [tex]\( r \)[/tex] is the annual percentage rate, and [tex]\( k \)[/tex] is the number of times each year that the interest is compounded.

(A) What values should be used for [tex]\( a, r \)[/tex], and [tex]\( k \)[/tex]?
[tex]\[ a = \square \][/tex]
[tex]\[ r = \square \][/tex]
[tex]\[ k = \square \][/tex]

(B) How much money will Manuela have in the account in 10 years?

Answer: [tex]\( \$ \square \)[/tex] (Round answer to the nearest penny.)

(C) What is the annual percentage yield (APY) for the savings account? (The APY is the actual or effective annual percentage rate which includes all compounding in the year).

[tex]\[ APY = \square \% \][/tex] (Round answer to 3 decimal places.)

Sagot :

GinnyAnswer
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]
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