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Answer :
To solve this problem, we want to determine the equation that represents how much money Josiah will have in his account after accruing interest over a period of years. Since the interest is compounded annually, we'll use the compound interest formula, which is:
[tex]\[ y = P(1 + r)^x \][/tex]
where:
- [tex]\( y \)[/tex] is the amount of money in the account after [tex]\( x \)[/tex] years,
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money invested),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal), and
- [tex]\( x \)[/tex] is the number of years.
Let's break it down:
1. Identify the principal amount ([tex]\( P \)[/tex]):
Josiah's initial investment is [tex]\( \$360 \)[/tex]. So, [tex]\( P = 360 \)[/tex].
2. Identify the annual interest rate ([tex]\( r \)[/tex]):
The account accrues interest at a rate of [tex]\( 3\% \)[/tex] annually. To use this in the formula, we convert the percentage to a decimal by dividing by 100, giving us [tex]\( r = 0.03 \)[/tex].
3. Write the equation using the formula:
Substitute the values of [tex]\( P \)[/tex] and [tex]\( r \)[/tex] into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
Simplifying [tex]\( (1 + 0.03) \)[/tex] gives us [tex]\( 1.03 \)[/tex]. So, the equation becomes:
[tex]\[
y = 360(1.03)^x
\][/tex]
Therefore, the equation that models the amount of money in Josiah's account after [tex]\( x \)[/tex] years, with no additional deposits or withdrawals, is:
[tex]\[ y = 360(1.03)^x \][/tex]
This represents the original amount plus the accrued interest over time.
[tex]\[ y = P(1 + r)^x \][/tex]
where:
- [tex]\( y \)[/tex] is the amount of money in the account after [tex]\( x \)[/tex] years,
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money invested),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal), and
- [tex]\( x \)[/tex] is the number of years.
Let's break it down:
1. Identify the principal amount ([tex]\( P \)[/tex]):
Josiah's initial investment is [tex]\( \$360 \)[/tex]. So, [tex]\( P = 360 \)[/tex].
2. Identify the annual interest rate ([tex]\( r \)[/tex]):
The account accrues interest at a rate of [tex]\( 3\% \)[/tex] annually. To use this in the formula, we convert the percentage to a decimal by dividing by 100, giving us [tex]\( r = 0.03 \)[/tex].
3. Write the equation using the formula:
Substitute the values of [tex]\( P \)[/tex] and [tex]\( r \)[/tex] into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
Simplifying [tex]\( (1 + 0.03) \)[/tex] gives us [tex]\( 1.03 \)[/tex]. So, the equation becomes:
[tex]\[
y = 360(1.03)^x
\][/tex]
Therefore, the equation that models the amount of money in Josiah's account after [tex]\( x \)[/tex] years, with no additional deposits or withdrawals, is:
[tex]\[ y = 360(1.03)^x \][/tex]
This represents the original amount plus the accrued interest over time.
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