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Answer :
- Identify the principal amount ($P = 360$) and the annual interest rate ($r = 0.03$).
- Substitute these values into the compound interest formula: $y = P(1 + r)^x$.
- Simplify the equation: $y = 360(1 + 0.03)^x = 360(1.03)^x$.
- The equation representing the amount of money in Josiah's account after $x$ years is $\boxed{y=360(1.03)^x}$.
### Explanation
1. Understanding the Problem
We are given an initial investment of $360 and an annual interest rate of 3%. We need to find the equation that represents the amount of money in the account after x years. The formula for compound interest is $A = P(1 + r)^t$, where A is the final amount, P is the principal amount, r is the interest rate, and t is the number of years.
2. Identifying the Values
The principal amount, $P = 360$. The annual interest rate, $r = 3\% = 0.03$. The number of years, $t = x$.
3. Applying the Formula
Substitute the values into the compound interest formula: $y = P(1 + r)^x$. Substitute $P = 360$ and $r = 0.03$ into the formula: $y = 360(1 + 0.03)^x$. Simplify the equation: $y = 360(1.03)^x$.
### Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how your investments grow over time with compound interest helps you estimate your potential savings. Similarly, when taking out a loan, understanding the compound interest allows you to calculate the total repayment amount and make better borrowing decisions. This concept is also applicable in business, where companies use compound interest calculations to forecast the growth of their investments and profits.
- Substitute these values into the compound interest formula: $y = P(1 + r)^x$.
- Simplify the equation: $y = 360(1 + 0.03)^x = 360(1.03)^x$.
- The equation representing the amount of money in Josiah's account after $x$ years is $\boxed{y=360(1.03)^x}$.
### Explanation
1. Understanding the Problem
We are given an initial investment of $360 and an annual interest rate of 3%. We need to find the equation that represents the amount of money in the account after x years. The formula for compound interest is $A = P(1 + r)^t$, where A is the final amount, P is the principal amount, r is the interest rate, and t is the number of years.
2. Identifying the Values
The principal amount, $P = 360$. The annual interest rate, $r = 3\% = 0.03$. The number of years, $t = x$.
3. Applying the Formula
Substitute the values into the compound interest formula: $y = P(1 + r)^x$. Substitute $P = 360$ and $r = 0.03$ into the formula: $y = 360(1 + 0.03)^x$. Simplify the equation: $y = 360(1.03)^x$.
### Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how your investments grow over time with compound interest helps you estimate your potential savings. Similarly, when taking out a loan, understanding the compound interest allows you to calculate the total repayment amount and make better borrowing decisions. This concept is also applicable in business, where companies use compound interest calculations to forecast the growth of their investments and profits.
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