A bank features a savings account that has an annual percentage rate of r=2.6% with interest compounded quarterly. Kyle deposits $8,500 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 Kyle have in the account in 7 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= %.

Let's go through each part of the problem step by step.

### (A) Determining the Values for $a$, $r$, and $k$

- $a$: The principal (starting amount) Kyle deposits, which is **$8,500**.
- $r$: The annual percentage rate as a decimal. Since the rate is **2.6%**, we convert it to a decimal by dividing by 100:  
  \[
  r = \frac{2.6}{100} = 0.026
  \]
- $k$: The number of times the interest is compounded annually. Since it is compounded **quarterly**, $k = 4$.

Thus:
- $a = 8500$
- $r = 0.026$
- $k = 4$

### (B) Calculating the Account Balance After 7 Years

The formula for the account balance after $t$ years is:

\[
A(t) = a \left(1 + \frac{r}{k}\right)^{kt}
\]

Substitute the values:
- $a = 8500$
- $r = 0.026$
- $k = 4$
- $t = 7$

Plug these into the formula:

\[
A(7) = 8500 \left(1 + \frac{0.026}{4}\right)^{4 \times 7}
\]

Let's calculate this.

The amount of money Kyle will have in the account after 7 years is approximately **$10,190.72**.

### (C) Calculating the Annual Percentage Yield (APY)

The Annual Percentage Yield (APY) is calculated using the formula:

\[
\text{APY} = \left(1 + \frac{r}{k}\right)^k - 1
\]

Substitute the values:
- $r = 0.026$
- $k = 4$

Let's calculate the APY.The Annual Percentage Yield (APY) for the savings account is approximately **2.63%**.

### Summary of Answers:
(A) Values:
- $a = 8500$
- $r = 0.026$
- $k = 4$

(B) Kyle will have **$10,190.72** in the account after 7 years.

(C) The APY for the savings account is **2.63%**.

Would you like further details or have any questions?

Here are some related questions to consider:
1. What would be the account balance if the interest were compounded monthly instead of quarterly?
2. How much interest would Kyle earn in total over the 7 years?
3. What would happen to the APY if the interest rate were increased to 3%?
4. How long would it take for Kyle’s initial deposit to double with this interest rate?
5. How would the APY change if the bank compounded interest daily instead of quarterly?

**Tip:** The APY provides a better understanding of the actual earnings on an investment compared to the nominal interest rate, especially when compounding occurs multiple times a year.

Explore how to calculate the growth of a savings account with compound interest using the exponential growth formula. Understand the impact of quarterly compounding on Kyle's $8,500 deposit over 7 years, including the annual percentage yield (APY) calculation. Round answers to the nearest penny for accuracy.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation