Amelia is going to invest in an account paying an interest rate of 2.3% compounded quarterly. How much would Amelia need to invest, to the nearest hundred dollars, for the value of the account to reach $1,530 in 19 years?
Amelia is going to invest in an account paying an interest rate of 2.3% compounded quarterly. How much would Amelia need to invest, to the nearest hundred dollars, for the value of the account to reach $1,530 in 19 years?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Investment Problem Solving**
Amelia plans to invest in an account that offers an interest rate of 2.3%, compounded quarterly. The goal is for the account balance to reach $1,530 in 19 years. The task is to determine the amount Amelia needs to invest initially, rounding to the nearest hundred dollars.
**Concepts:**
- **Compounded Interest**: Refers to the interest calculated on the initial principal, which also includes all the accumulated interest from previous periods.
- **Quarterly Compounding**: Interest is compounded four times a year.
**Formula Used**:
To solve this, we can use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the number of years the money is invested for.
Given:
- \( A = 1530 \)
- \( r = 0.023 \)
- \( n = 4 \)
- \( t = 19 \)
**Calculate \( P \)**:
Rearrange the formula to find the principal \( P \):
\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]
Plug in the values to calculate the initial investment required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08395658-3915-4b41-a901-f1ebc73d92aa%2F4c0309f2-555e-4b6c-805c-c396c576db23%2Fxrfdaig_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Investment Problem Solving**
Amelia plans to invest in an account that offers an interest rate of 2.3%, compounded quarterly. The goal is for the account balance to reach $1,530 in 19 years. The task is to determine the amount Amelia needs to invest initially, rounding to the nearest hundred dollars.
**Concepts:**
- **Compounded Interest**: Refers to the interest calculated on the initial principal, which also includes all the accumulated interest from previous periods.
- **Quarterly Compounding**: Interest is compounded four times a year.
**Formula Used**:
To solve this, we can use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the number of years the money is invested for.
Given:
- \( A = 1530 \)
- \( r = 0.023 \)
- \( n = 4 \)
- \( t = 19 \)
**Calculate \( P \)**:
Rearrange the formula to find the principal \( P \):
\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]
Plug in the values to calculate the initial investment required.
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