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Answer :
To find the equation representing the amount of money in Josiah's account after a certain number of years, we will use the compound interest formula. Here’s how we can set it up:
1. Understand the terms:
- Josiah's initial investment, or principal amount, is \$360.
- The annual interest rate is 3%. In decimal form, that's 0.03.
2. Compound Interest Formula:
The formula to determine the amount of money in an account with compound interest is:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [tex]\(y\)[/tex] is the final amount in the account after [tex]\(x\)[/tex] years,
- [tex]\(P\)[/tex] is the principal amount (initial investment),
- [tex]\(r\)[/tex] is the annual interest rate (as a decimal),
- [tex]\(x\)[/tex] is the number of years the money is invested.
3. Setting up the equation for Josiah's account:
- Substitute [tex]\(P = 360\)[/tex] and [tex]\(r = 0.03\)[/tex] into the formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- Simplify inside the parentheses:
[tex]\[
y = 360(1.03)^x
\][/tex]
Thus, the equation that represents the amount of money in Josiah's account after [tex]\(x\)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
This equation correctly models the growth of the investment due to annual compound interest.
1. Understand the terms:
- Josiah's initial investment, or principal amount, is \$360.
- The annual interest rate is 3%. In decimal form, that's 0.03.
2. Compound Interest Formula:
The formula to determine the amount of money in an account with compound interest is:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [tex]\(y\)[/tex] is the final amount in the account after [tex]\(x\)[/tex] years,
- [tex]\(P\)[/tex] is the principal amount (initial investment),
- [tex]\(r\)[/tex] is the annual interest rate (as a decimal),
- [tex]\(x\)[/tex] is the number of years the money is invested.
3. Setting up the equation for Josiah's account:
- Substitute [tex]\(P = 360\)[/tex] and [tex]\(r = 0.03\)[/tex] into the formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- Simplify inside the parentheses:
[tex]\[
y = 360(1.03)^x
\][/tex]
Thus, the equation that represents the amount of money in Josiah's account after [tex]\(x\)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
This equation correctly models the growth of the investment due to annual compound interest.
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