pre calc question

Question

A bank features a savings account that has an annual percentage rate of r=3.2% with interest compounded quarterly. Pilar deposits $11,000 into the account. 

The account balance can be modeled by the exponential formula A(t)=a(1+rk)kt, 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? 

a= , r= , k= 

(B) How much money will Pilar have in the account in 8 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). 

APY= %. 

Round answer to 3 decimal places.

Answer 1

A) Principal amount (starting amount) = $11,000, which is the amount that Pilar deposits into the account.

Therefore, a = $11,000.

Annual percentage rate (APR) = 3.2%. Since the interest is compounded quarterly, we need to divide the annual percentage rate by the number of compounding periods per year (k) to get the quarterly rate.

Therefore, we have:

1. k = 4 (quarterly compounding)
2. r = 3.2% / k = 0.8% (quarterly rate)

So the values to be used in the formula A(t)=a(1+rk)^kt are: a = $11,000, r = 0.8%, and k = 4.

B) To find the account balance after 8 years, we can use the exponential formula:

A(t) = a(1 + r/k)^(kt)

where:

1. a = $11,000 (the principal amount)
2. r = 3.2% (the annual percentage rate)
3. k = 4 (the number of times per year that interest is compounded)
4. t = 8 (the number of years)

Plugging in these values, we get:

A(8) = $11,000(1 + 0.032/4)^(4*8) = $11,000(1 + 0.008)^32 = $11,000(1.008)^32 = $14,527.81 (rounded to the nearest cent)

Therefore, after 8 years, Pilar will have $14,527.81 in the account.

C) The annual percentage yield (APY) is the actual or effective annual percentage rate that takes into account the effect of compounding.

To calculate the APY for the savings account with an annual percentage rate of 3.2% compounded quarterly, we can use the following formula:

APY = (1 + r/k)^k - 1

where:

1. r = 3.2% (the annual percentage rate)
2. k = 4 (the number of times per year that interest is compounded)

Plugging in these values, we get:

APY = (1 + 0.032/4)^4 - 1 = 0.0323 or 3.23%

Thus, the savings account's annual percentage yield (APY) is 3.23%.