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Answer :
To solve this problem, let's understand how interest accrual works. Josiah has invested \[tex]$360 in an account that earns 3% interest annually. We want to find an equation that represents the amount of money in the account, \( y \), after \( x \) years.
Here's a step-by-step solution:
1. Initial Investment: Josiah starts with \(\$[/tex]360\).
2. Interest Rate: The annual interest rate is [tex]\(3\%\)[/tex].
3. Compound Interest Formula: For compound interest, the formula to calculate the future amount after [tex]\( x \)[/tex] years is:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [tex]\( P \)[/tex] is the initial investment (\$360 in this case),
- [tex]\( r \)[/tex] is the interest rate (expressed as a decimal, so 3% becomes 0.03),
- [tex]\( x \)[/tex] is the number of years.
4. Applying the Formula: Plug the values into the formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
5. Simplify:
[tex]\[
y = 360(1.03)^x
\][/tex]
Therefore, the correct equation representing the amount of money in Josiah's account after [tex]\( x \)[/tex] years is [tex]\( y = 360(1.03)^x \)[/tex].
This matches the option [tex]\( y = 360(1.03)^x \)[/tex], which is the correct choice from the given options.
Here's a step-by-step solution:
1. Initial Investment: Josiah starts with \(\$[/tex]360\).
2. Interest Rate: The annual interest rate is [tex]\(3\%\)[/tex].
3. Compound Interest Formula: For compound interest, the formula to calculate the future amount after [tex]\( x \)[/tex] years is:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [tex]\( P \)[/tex] is the initial investment (\$360 in this case),
- [tex]\( r \)[/tex] is the interest rate (expressed as a decimal, so 3% becomes 0.03),
- [tex]\( x \)[/tex] is the number of years.
4. Applying the Formula: Plug the values into the formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
5. Simplify:
[tex]\[
y = 360(1.03)^x
\][/tex]
Therefore, the correct equation representing the amount of money in Josiah's account after [tex]\( x \)[/tex] years is [tex]\( y = 360(1.03)^x \)[/tex].
This matches the option [tex]\( y = 360(1.03)^x \)[/tex], which is the correct choice from the given options.
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