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Answer :
Sure! Let's solve the problem step-by-step to find the correct equation representing the amount of money in Josiah's account after [tex]\( x \)[/tex] years.
Josiah invests \[tex]$360 in an account that accrues 3% interest annually. We need to determine which equation correctly expresses the amount of money, \( y \), in Josiah's account after \( x \) years, with no additional deposits or withdrawals.
1. Understanding Compound Interest:
- When interest is compounded annually, the formula to calculate the amount of money in the account after \( x \) years is:
\[
A = P(1 + r)^x
\]
- Where:
- \( A \) is the amount of money accumulated after \( x \) years, including interest.
- \( P \) is the principal amount (initial amount invested), which is \$[/tex]360.
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal). Since the interest rate is 3%, [tex]\( r = 0.03 \)[/tex].
- [tex]\( x \)[/tex] is the number of years the money is invested.
2. Substitute the Given Values:
- Given:
[tex]\[
P = 360
\][/tex]
[tex]\[
r = 0.03
\][/tex]
- Plug these values into the formula:
[tex]\[
A = 360(1 + 0.03)^x
\][/tex]
- Simplify inside the parentheses:
[tex]\[
A = 360(1.03)^x
\][/tex]
3. Identify the Correct Equation:
- The simplified equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
So, the correct equation is:
[tex]\[ y = 360(1.03)^x \][/tex]
This matches the last option provided in the question.
Josiah invests \[tex]$360 in an account that accrues 3% interest annually. We need to determine which equation correctly expresses the amount of money, \( y \), in Josiah's account after \( x \) years, with no additional deposits or withdrawals.
1. Understanding Compound Interest:
- When interest is compounded annually, the formula to calculate the amount of money in the account after \( x \) years is:
\[
A = P(1 + r)^x
\]
- Where:
- \( A \) is the amount of money accumulated after \( x \) years, including interest.
- \( P \) is the principal amount (initial amount invested), which is \$[/tex]360.
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal). Since the interest rate is 3%, [tex]\( r = 0.03 \)[/tex].
- [tex]\( x \)[/tex] is the number of years the money is invested.
2. Substitute the Given Values:
- Given:
[tex]\[
P = 360
\][/tex]
[tex]\[
r = 0.03
\][/tex]
- Plug these values into the formula:
[tex]\[
A = 360(1 + 0.03)^x
\][/tex]
- Simplify inside the parentheses:
[tex]\[
A = 360(1.03)^x
\][/tex]
3. Identify the Correct Equation:
- The simplified equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
So, the correct equation is:
[tex]\[ y = 360(1.03)^x \][/tex]
This matches the last option provided in the question.
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