Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter4: Exponential And Logarithmic Functions
Section4.CT: Chapter Test
Problem 11CT
Related questions
Question
![**Compound Interest Calculation Example**
**Question:**
If Frank invests $11,400 at 3.7% interest compounded quarterly, how much will his investment be worth in 9 years?
**Explanation:**
To find the future value of an investment using compound interest, we use the formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money), which is \$11,400.
- \( r \) is the annual interest rate (decimal), so 3.7% becomes 0.037.
- \( n \) is the number of times that interest is compounded per year. In this case, it is compounded quarterly, so \( n = 4 \).
- \( t \) is the time the money is invested for in years, which is 9 years.
First, convert the annual interest rate from a percentage to a decimal:
\[ r = \frac{3.7}{100} = 0.037 \]
Next, apply the values to the formula:
\[ A = 11400 \left(1 + \frac{0.037}{4}\right)^{4 \times 9} \]
Calculate the value inside the parentheses:
\[ 1 + \frac{0.037}{4} = 1 + 0.00925 = 1.00925 \]
Raise 1.00925 to the power of \( 4 \times 9 \) (which is 36):
\[ 1.00925^{36} \approx 1.397 \]
Finally, multiply this result by the principal amount:
\[ A = 11400 \times 1.397 \approx 15972.3 \]
Thus, after 9 years, Frank’s investment will be worth approximately \$15,972.30.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9168bf64-9e72-49b9-9a77-a0f13a9541a3%2F269ed15d-915e-4e2b-bc02-33ad43a6a2bb%2Fs7onm19_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Compound Interest Calculation Example**
**Question:**
If Frank invests $11,400 at 3.7% interest compounded quarterly, how much will his investment be worth in 9 years?
**Explanation:**
To find the future value of an investment using compound interest, we use the formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money), which is \$11,400.
- \( r \) is the annual interest rate (decimal), so 3.7% becomes 0.037.
- \( n \) is the number of times that interest is compounded per year. In this case, it is compounded quarterly, so \( n = 4 \).
- \( t \) is the time the money is invested for in years, which is 9 years.
First, convert the annual interest rate from a percentage to a decimal:
\[ r = \frac{3.7}{100} = 0.037 \]
Next, apply the values to the formula:
\[ A = 11400 \left(1 + \frac{0.037}{4}\right)^{4 \times 9} \]
Calculate the value inside the parentheses:
\[ 1 + \frac{0.037}{4} = 1 + 0.00925 = 1.00925 \]
Raise 1.00925 to the power of \( 4 \times 9 \) (which is 36):
\[ 1.00925^{36} \approx 1.397 \]
Finally, multiply this result by the principal amount:
\[ A = 11400 \times 1.397 \approx 15972.3 \]
Thus, after 9 years, Frank’s investment will be worth approximately \$15,972.30.
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