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Answer :
To solve this problem, we need to understand how interest compounds annually in a bank account. The key is to use the formula for compound interest, 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 (initial investment).
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( x \)[/tex] is the number of years the money is invested.
Let's break it down:
1. Principal (Initial Amount): Josiah invests \$360, so [tex]\( P = 360 \)[/tex].
2. Interest Rate: The account has an interest rate of 3% annually. To use this in our formula, we need to express the percentage as a decimal. Therefore, 3% becomes 0.03, so [tex]\( r = 0.03 \)[/tex].
3. Compounded Annually: Since the problem states the interest compounds annually and no deposits or withdrawals are made, we use the compound interest formula as shown above.
Plugging these values into the compound interest formula gives:
[tex]\[ y = 360(1 + 0.03)^x \][/tex]
Simplifying inside the parentheses:
[tex]\[ y = 360(1.03)^x \][/tex]
This equation [tex]\( y = 360(1.03)^x \)[/tex] represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years, considering the conditions given with an annual compounding interest of 3%.
So, the correct equation from the given options is:
[tex]\[ y = 360(1.03)^x \][/tex]
[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 (initial investment).
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( x \)[/tex] is the number of years the money is invested.
Let's break it down:
1. Principal (Initial Amount): Josiah invests \$360, so [tex]\( P = 360 \)[/tex].
2. Interest Rate: The account has an interest rate of 3% annually. To use this in our formula, we need to express the percentage as a decimal. Therefore, 3% becomes 0.03, so [tex]\( r = 0.03 \)[/tex].
3. Compounded Annually: Since the problem states the interest compounds annually and no deposits or withdrawals are made, we use the compound interest formula as shown above.
Plugging these values into the compound interest formula gives:
[tex]\[ y = 360(1 + 0.03)^x \][/tex]
Simplifying inside the parentheses:
[tex]\[ y = 360(1.03)^x \][/tex]
This equation [tex]\( y = 360(1.03)^x \)[/tex] represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years, considering the conditions given with an annual compounding interest of 3%.
So, the correct equation from the given options is:
[tex]\[ y = 360(1.03)^x \][/tex]
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