A bank features a savings account that has an annual percentage rate of
r
=
4.2
% with interest compounded quarterly. Connor deposits $6,000 into the account.
The account balance can be modeled by the exponential formula
A
(
t
)
=
a
(
1
+
r
k
)
k
t
, where
A
is account value after
t
years ,
a
is the principal (starting amount),
r
is the annual percentage rate,
k
is the number of times each year that the interest is compounded.
(A) What values should be used for
a
, r, and
k
?
(B) How much money will Connor have in the account in
9
years?
Answer = $
.
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).
A
P
Y
=
%.
(A) The value for a, r, and k should be:
a = $6,000
r = 4.2%
k = 4 (since the interest is compounded quarterly)
(B) To find how much money Connor will have in the account in 9 years, we can use the formula A(t) = a(1 + r/k)kt and substitute in the values:
A(9) = 6000(1 + 0.042/4)4(9)
A(9) = 6000(1 + 0.0105)^36
A(9) ≈ $8,238.55
Therefore, Connor will have approximately $8,238.55 in the account after 9 years.
(C) The annual percentage yield (APY) is calculated using the formula APY = (1 + r/k)^k - 1 and substituting in the values:
APY = (1 + 0.042/4)^4 - 1
APY = (1 + 0.0105)^4 - 1
APY ≈ 0.0437
The annual percentage yield (APY) for the savings account is approximately 4.37%.
(A) The values for a, r, and k are as follows:
a = $6,000 (Connor's initial deposit into the account)
r = 4.2% (annual percentage rate)
k = 4 (interest compounded quarterly, so 4 times per year)
(B) To calculate the amount of money Connor will have in the account after 9 years, we can use the formula:
A(t) = a(1 + r/k)^(kt)
Substituting the given values:
A(9) = 6,000(1 + 0.042/4)^(4*9)
Calculate the expression inside the parentheses first:
1 + 0.042/4 = 1 + 0.0105 = 1.0105
Then:
A(9) = 6,000(1.0105)^(4*9)
A(9) ≈ $8,223.49 (rounded to the nearest penny)
Therefore, Connor will have approximately $8,223.49 in the account after 9 years.
(C) The annual percentage yield (APY) can be calculated using the formula:
APY = (1 + r/k)^k - 1
Substituting the given values:
APY = (1 + 0.042/4)^4 - 1
APY ≈ 0.0431
Therefore, the annual percentage yield for the savings account is approximately 4.31%.