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Josiah invests [tex]\$360[/tex] into an account that accrues [tex]3\%[/tex] interest annually. Assuming no deposits or withdrawals are made, which equation represents the amount of money in Josiah's account, [tex]y[/tex], after [tex]x[/tex] years?

A. [tex]y = 360(1.3)^x[/tex]
B. [tex]y = 360(0.3)^x[/tex]
C. [tex]y = 360(0.03)^x[/tex]
D. [tex]y = 360(1.03)^x[/tex]

Answer :

To find the equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years with an annual interest rate of 3%, we follow these steps:

1. Understand Compound Interest:
The formula for compound interest, which is applied annually, is:
[tex]\[
A = P(1 + r)^x
\][/tex]

Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( x \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( x \)[/tex] is the number of years the money is invested for.

2. Identify the Values:
- The principal amount [tex]\( P \)[/tex] is $360.
- The annual interest rate [tex]\( r \)[/tex] is 3%, which is 0.03 as a decimal.

3. Substitute the Values into the Formula:
Plug in [tex]\( P = 360 \)[/tex] and [tex]\( r = 0.03 \)[/tex] into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]

4. Simplify the Equation:
Simplify the expression inside the parentheses:
[tex]\[
1 + 0.03 = 1.03
\][/tex]

So, the equation becomes:
[tex]\[
y = 360(1.03)^x
\][/tex]

5. Select the Correct Option:
After simplifying, we see that the correct equation from the provided options is:
[tex]\[
y = 360(1.03)^x
\][/tex]

This equation correctly represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years of compounding annually at a 3% interest rate.

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