If $500 is invested at an interest rate of 3.5% per year, find the amount of the investment at the end of 11 years for the following compounding methods. (Round your answers to the nearest cent.)
(a) Annually
$ 
 
Incorrect: Your answer is incorrect.


(b) Semiannually
$ 
 
Incorrect: Your answer is incorrect.


(c) Quarterly
$ 
 
Incorrect: Your answer is incorrect.


(d) Continuousl

To solve this problem, we use the following formulas for compound interest depending on the method of compounding:

### General Compound Interest Formula (for non-continuous compounding):
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ A $ is the amount of the investment after $ t $ years.
- $ P $ is the principal (initial investment).
- $ r $ is the annual interest rate (in decimal form).
- $ n $ is the number of times interest is compounded per year.
- $ t $ is the number of years.

### Continuous Compounding Formula:
\[
A = P e^{rt}
\]
Where:
- $ e $ is Euler's number (approximately 2.71828).

Given:
- $ P = 500 $ (the initial investment)
- $ r = 3.5\% = 0.035 $ (interest rate in decimal form)
- $ t = 11 $ years

Let's calculate the investment amount for each method:

### (a) Compounded Annually:
For annual compounding, $ n = 1 $.
\[
A = 500 \left(1 + \frac{0.035}{1}\right)^{1 \times 11} = 500 (1.035)^{11}
\]

### (b) Compounded Semiannually:
For semiannual compounding, $ n = 2 $.
\[
A = 500 \left(1 + \frac{0.035}{2}\right)^{2 \times 11} = 500 \left(1 + 0.0175\right)^{22}
\]

### (c) Compounded Quarterly:
For quarterly compounding, $ n = 4 $.
\[
A = 500 \left(1 + \frac{0.035}{4}\right)^{4 \times 11} = 500 \left(1 + 0.00875\right)^{44}
\]

### (d) Compounded Continuously:
For continuous compounding, use the continuous formula:
\[
A = 500 e^{0.035 \times 11}
\]

I'll calculate these values for you.The amount of the investment at the end of 11 years, depending on the compounding method, is as follows:

(a) Compounded annually: **$729.98**

(b) Compounded semiannually: **$732.36**

(c) Compounded quarterly: **$733.58**

(d) Compounded continuously: **$734.81**

Would you like any further details or explanations on how these were calculated?

Here are 5 related questions for deeper exploration:
1. What would the final amount be if compounded monthly?
2. How does the compounding frequency affect the final investment?
3. How would the investment change with a different interest rate, say 5%?
4. What happens to the final amount if the time period is extended to 20 years?
5. How does continuous compounding compare with quarterly compounding over shorter time spans?

**Tip:** The more frequently interest is compounded, the higher the final amount will be due to the effect of interest on interest.

If $500 is invested at an interest rate of 3.5% per year, find the amount of the investment at the end of 11 years for the following compounding methods: (a) Annually, (b) Semiannually, (c) Quarterly, (d) Continuously.

Solve a compound interest problem where $500 is invested at 3.5% per year over 11 years with different compounding methods. Learn how to calculate the final amount for annual, semiannual, quarterly, and continuous compounding. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation