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Answer :
To determine the equation that represents the amount of money in Josiah's account after earning interest annually, let's break it down step-by-step:
1. Initial Investment and Interest Rate:
- Josiah initially invests \$360.
- The annual interest rate is [tex]\(3\%\)[/tex]. In decimal form, this rate is 0.03.
2. Understanding Compound Interest:
- Compound interest means that the interest you earn each year is added to your principal, forming a new, larger principal amount for the next year's interest calculation.
3. Compound Interest Formula:
- The general formula for compound interest is [tex]\(y = P(1 + r)^x\)[/tex], where:
- [tex]\(P\)[/tex] is the principal amount (initial investment).
- [tex]\(r\)[/tex] is the annual interest rate in decimal form.
- [tex]\(x\)[/tex] is the number of years.
- [tex]\(y\)[/tex] is the amount of money after [tex]\(x\)[/tex] years.
4. Applying the Values to the Formula:
- Here, the principal [tex]\(P = 360\)[/tex].
- The interest rate [tex]\(r = 0.03\)[/tex].
- Substitute these values into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- Simplifying the expression inside the parentheses:
[tex]\[
y = 360(1.03)^x
\][/tex]
This simplified equation, [tex]\(y = 360(1.03)^x\)[/tex], accurately represents the amount of money in Josiah's account, factoring in the compound interest over [tex]\(x\)[/tex] years. Therefore, the correct equation is [tex]\(y = 360(1.03)^x\)[/tex].
1. Initial Investment and Interest Rate:
- Josiah initially invests \$360.
- The annual interest rate is [tex]\(3\%\)[/tex]. In decimal form, this rate is 0.03.
2. Understanding Compound Interest:
- Compound interest means that the interest you earn each year is added to your principal, forming a new, larger principal amount for the next year's interest calculation.
3. Compound Interest Formula:
- The general formula for compound interest is [tex]\(y = P(1 + r)^x\)[/tex], where:
- [tex]\(P\)[/tex] is the principal amount (initial investment).
- [tex]\(r\)[/tex] is the annual interest rate in decimal form.
- [tex]\(x\)[/tex] is the number of years.
- [tex]\(y\)[/tex] is the amount of money after [tex]\(x\)[/tex] years.
4. Applying the Values to the Formula:
- Here, the principal [tex]\(P = 360\)[/tex].
- The interest rate [tex]\(r = 0.03\)[/tex].
- Substitute these values into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- Simplifying the expression inside the parentheses:
[tex]\[
y = 360(1.03)^x
\][/tex]
This simplified equation, [tex]\(y = 360(1.03)^x\)[/tex], accurately represents the amount of money in Josiah's account, factoring in the compound interest over [tex]\(x\)[/tex] years. Therefore, the correct equation is [tex]\(y = 360(1.03)^x\)[/tex].
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