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Answer :
To determine the correct equation for the amount of money in Josiah's account after [tex]\( x \)[/tex] years, we'll consider an important concept in finance called compound interest. Here's how we arrive at the solution step-by-step:
1. Understand the Initial Investment and Interest Rate:
- Josiah initially invests [tex]$360 into the account.
- The account gains an interest of \( 3\% \) annually.
2. Identify the Formula for Compound Interest:
The general formula for an account with compound interest is:
\[
y = P \times (1 + r)^x
\]
where:
- \( y \) is the amount of money after \( x \) years,
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate in decimal form,
- \( x \) is the number of years the money is invested.
3. Apply the Given Values to the Formula:
- \( P = 360 \) (since he invests $[/tex]360 initially),
- [tex]\( r = 0.03 \)[/tex] (the interest rate is [tex]\( 3\% \)[/tex], which is [tex]\( 0.03 \)[/tex] in decimal form).
4. Substitute the Values into the Formula:
- Replace [tex]\( P \)[/tex] and [tex]\( r \)[/tex] in the formula:
[tex]\[
y = 360 \times (1 + 0.03)^x
\][/tex]
5. Simplify the Equation:
- This becomes:
[tex]\[
y = 360 \times (1.03)^x
\][/tex]
Thus, the equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years, assuming no additional deposits or withdrawals, is:
[tex]\[
y = 360(1.03)^x
\][/tex]
That's why the correct choice is [tex]\( y = 360(1.03)^x \)[/tex].
1. Understand the Initial Investment and Interest Rate:
- Josiah initially invests [tex]$360 into the account.
- The account gains an interest of \( 3\% \) annually.
2. Identify the Formula for Compound Interest:
The general formula for an account with compound interest is:
\[
y = P \times (1 + r)^x
\]
where:
- \( y \) is the amount of money after \( x \) years,
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate in decimal form,
- \( x \) is the number of years the money is invested.
3. Apply the Given Values to the Formula:
- \( P = 360 \) (since he invests $[/tex]360 initially),
- [tex]\( r = 0.03 \)[/tex] (the interest rate is [tex]\( 3\% \)[/tex], which is [tex]\( 0.03 \)[/tex] in decimal form).
4. Substitute the Values into the Formula:
- Replace [tex]\( P \)[/tex] and [tex]\( r \)[/tex] in the formula:
[tex]\[
y = 360 \times (1 + 0.03)^x
\][/tex]
5. Simplify the Equation:
- This becomes:
[tex]\[
y = 360 \times (1.03)^x
\][/tex]
Thus, the equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years, assuming no additional deposits or withdrawals, is:
[tex]\[
y = 360(1.03)^x
\][/tex]
That's why the correct choice is [tex]\( y = 360(1.03)^x \)[/tex].
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