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Answer :
To solve this problem, we're trying to determine which equation represents the amount of money in Josiah's account after a certain number of years, given a principal investment and an annual interest rate.
1. Understand the Compound Interest Formula:
- The formula to calculate the future value of an investment compounded annually is given by:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [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, and
- [tex]\( y \)[/tex] is the amount of money in the account after [tex]\( x \)[/tex] years.
2. Apply the Given Values:
- Here, the principal [tex]\( P \)[/tex] is \$360.
- The annual interest rate [tex]\( r \)[/tex] is 3%, or 0.03 as a decimal.
3. Substitute the Values Into the Formula:
- Using the formula above, we substitute the values:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- This simplifies to:
[tex]\[
y = 360(1.03)^x
\][/tex]
4. Choose the Correct Equation:
- From the given options, the equation that matches [tex]\( y = 360(1.03)^x \)[/tex] is the last one:
[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, with no additional deposits or withdrawals and compounded annually at a rate of 3%, is:
[tex]\[ y = 360(1.03)^x \][/tex]
1. Understand the Compound Interest Formula:
- The formula to calculate the future value of an investment compounded annually is given by:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [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, and
- [tex]\( y \)[/tex] is the amount of money in the account after [tex]\( x \)[/tex] years.
2. Apply the Given Values:
- Here, the principal [tex]\( P \)[/tex] is \$360.
- The annual interest rate [tex]\( r \)[/tex] is 3%, or 0.03 as a decimal.
3. Substitute the Values Into the Formula:
- Using the formula above, we substitute the values:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- This simplifies to:
[tex]\[
y = 360(1.03)^x
\][/tex]
4. Choose the Correct Equation:
- From the given options, the equation that matches [tex]\( y = 360(1.03)^x \)[/tex] is the last one:
[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, with no additional deposits or withdrawals and compounded annually at a rate of 3%, is:
[tex]\[ y = 360(1.03)^x \][/tex]
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