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Answer :
To solve the problem of finding the equation that represents the amount of money in Josiah's account after [tex]$x$[/tex] years, we'll use the formula for compound interest.
### Step-by-Step Solution:
1. Understand the Compound Interest Formula:
The general formula to calculate compound interest is:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [tex]\( y \)[/tex] is the amount of money accumulated after [tex]\( x \)[/tex] years, including interest.
- [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 the money is invested for.
2. Identify the Given Values:
- Josiah's initial investment [tex]\( P \)[/tex] is $360.
- The annual interest rate is 3%. To convert this percentage into a decimal, divide by 100:
[tex]\[
r = \frac{3}{100} = 0.03
\][/tex]
3. Substitute the Known Values into the Formula:
Replace [tex]\( P \)[/tex] with 360 and [tex]\( r \)[/tex] with 0.03 in the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
4. Simplify the Equation:
Add 1 and the interest rate in the formula:
[tex]\[
y = 360(1.03)^x
\][/tex]
Therefore, the correct 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 shows that each year, the amount in the account is multiplied by 1.03, reflecting the 3% annual growth due to interest.
### Step-by-Step Solution:
1. Understand the Compound Interest Formula:
The general formula to calculate compound interest is:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [tex]\( y \)[/tex] is the amount of money accumulated after [tex]\( x \)[/tex] years, including interest.
- [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 the money is invested for.
2. Identify the Given Values:
- Josiah's initial investment [tex]\( P \)[/tex] is $360.
- The annual interest rate is 3%. To convert this percentage into a decimal, divide by 100:
[tex]\[
r = \frac{3}{100} = 0.03
\][/tex]
3. Substitute the Known Values into the Formula:
Replace [tex]\( P \)[/tex] with 360 and [tex]\( r \)[/tex] with 0.03 in the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
4. Simplify the Equation:
Add 1 and the interest rate in the formula:
[tex]\[
y = 360(1.03)^x
\][/tex]
Therefore, the correct 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 shows that each year, the amount in the account is multiplied by 1.03, reflecting the 3% annual growth due to interest.
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