1.2.3 Quiz: Exponential FunctionsQuestion 10 Of 10How Much Would \$200 Invested At 6\% Interest, Compounded Annually, Be Worth After 6 Years? Round Your Answer To The Nearest Cent.$A(t) = P\left(1+\frac{r}{n}\right)^{n T}$A. \$286.67 B.
Question 10 of 10
How much would $200 invested at 6% interest, compounded annually, be worth after 6 years? Round your answer to the nearest cent.
Understanding Exponential Functions
Exponential functions are a fundamental concept in mathematics, particularly in finance and economics. They describe the growth or decay of a quantity over time, often modeled using the formula:
A(t) = P(1 + r/n)^(nt)
Where:
- A(t) is the amount of money after time t
- P is the principal amount (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 time the money is invested for, in years
Calculating the Future Value of an Investment
To calculate the future value of an investment, we need to plug in the given values into the formula:
- P = $200 (initial investment)
- r = 6% = 0.06 (annual interest rate)
- n = 1 (compounded annually)
- t = 6 years (time the money is invested for)
Substituting these values into the formula, we get:
A(6) = 200(1 + 0.06/1)^(1*6)
A(6) = 200(1 + 0.06)^6
A(6) = 200(1.06)^6
A(6) = 200 * 1.503635
A(6) = 300.773
Rounding to the nearest cent, we get:
A(6) ≈ $300.77
Conclusion
Therefore, $200 invested at 6% interest, compounded annually, would be worth approximately $300.77 after 6 years.
Key Takeaways
- Exponential functions describe the growth or decay of a quantity over time.
- The formula for calculating the future value of an investment is A(t) = P(1 + r/n)^(nt).
- To calculate the future value of an investment, we need to plug in the given values into the formula.
- Compounding interest annually can significantly increase the future value of an investment.
Practice Problems
- A person invests $1,000 at 4% interest, compounded annually, for 5 years. How much will the investment be worth after 5 years?
- A company invests $50,000 at 8% interest, compounded annually, for 10 years. How much will the investment be worth after 10 years?
- A person invests $500 at 2% interest, compounded annually, for 3 years. How much will the investment be worth after 3 years?
Solutions
- A(5) = 1000(1 + 0.04/1)^(1*5) ≈ $1,216.16
- A(10) = 50000(1 + 0.08/1)^(1*10) ≈ $86,951.32
- A(3) = 500(1 + 0.02/1)^(1*3) ≈ $555.12
1.2.3 Quiz: Exponential Functions =====================================
Question 10 of 10
How much would $200 invested at 6% interest, compounded annually, be worth after 6 years? Round your answer to the nearest cent.
Q&A: Exponential Functions
Q: What is an exponential function?
A: An exponential function is a mathematical function that describes the growth or decay of a quantity over time. It is often modeled using the formula A(t) = P(1 + r/n)^(nt), where A(t) is the amount of money after time t, P is the principal amount (initial investment), r is the annual interest rate (in decimal form), n is the number of times interest is compounded per year, and t is the time the money is invested for, in years.
Q: What is the formula for calculating the future value of an investment?
A: The formula for calculating the future value of an investment is A(t) = P(1 + r/n)^(nt), where A(t) is the amount of money after time t, P is the principal amount (initial investment), r is the annual interest rate (in decimal form), n is the number of times interest is compounded per year, and t is the time the money is invested for, in years.
Q: What is the difference between simple interest and compound interest?
A: Simple interest is calculated as a percentage of the principal amount, while compound interest is calculated as a percentage of the principal amount plus any accrued interest. Compound interest can significantly increase the future value of an investment over time.
Q: How often is interest compounded?
A: Interest can be compounded annually, semi-annually, quarterly, or monthly, depending on the investment and the financial institution.
Q: What is the effect of compounding interest on the future value of an investment?
A: Compounding interest can significantly increase the future value of an investment over time. The more frequently interest is compounded, the greater the effect on the future value of the investment.
Q: How can I calculate the future value of an investment using an exponential function?
A: To calculate the future value of an investment using an exponential function, you need to plug in the given values into the formula A(t) = P(1 + r/n)^(nt), where A(t) is the amount of money after time t, P is the principal amount (initial investment), r is the annual interest rate (in decimal form), n is the number of times interest is compounded per year, and t is the time the money is invested for, in years.
Q: What are some common applications of exponential functions in finance?
A: Exponential functions are commonly used in finance to calculate the future value of investments, determine the present value of future cash flows, and model the growth of populations and economies.
Q: How can I use exponential functions to make informed investment decisions?
A: You can use exponential functions to calculate the future value of investments, determine the present value of future cash flows, and model the growth of populations and economies. This can help you make informed investment decisions and achieve your financial goals.
Practice Problems
- A person invests $1,000 at 4% interest, compounded annually, for 5 years. How much will the investment be worth after 5 years?
- A company invests $50,000 at 8% interest, compounded annually, for 10 years. How much will the investment be worth after 10 years?
- A person invests $500 at 2% interest, compounded annually, for 3 years. How much will the investment be worth after 3 years?
Solutions
- A(5) = 1000(1 + 0.04/1)^(1*5) ≈ $1,216.16
- A(10) = 50000(1 + 0.08/1)^(1*10) ≈ $86,951.32
- A(3) = 500(1 + 0.02/1)^(1*3) ≈ $555.12