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Answer :
- The problem involves finding the equation for compound interest.
- The formula for compound interest is $A = P(1 + r)^t$.
- Substitute the given values: $P = 360$ and $r = 0.03$.
- The equation representing the amount of money 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, assuming no deposits or withdrawals are made.
2. Recalling the Compound Interest Formula
The formula for compound interest is given by:
$$A = P(1 + r)^t$$
where:
- A is the final amount
- P is the principal amount (initial investment)
- r is the interest rate (as a decimal)
- t is the number of years
3. Identifying the Given Values
In this problem, we have:
- P = $360
- r = 3% = 0.03
- t = x (number of years)
- A = y (amount of money after x years)
4. Applying the Formula and Simplifying
Substituting these values into the compound interest formula, we get:
$$y = 360(1 + 0.03)^x$$
Simplifying the equation:
$$y = 360(1.03)^x$$
5. Final Answer
Therefore, the equation that represents the amount of money in Josiah's account after x years is:
$$y = 360(1.03)^x$$
### Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, you can use this formula to estimate how much your investments will grow over time. Similarly, when taking out a loan, understanding compound interest helps you calculate the total amount you'll need to repay. This knowledge empowers you to make sound choices about saving, investing, and borrowing money.
- The formula for compound interest is $A = P(1 + r)^t$.
- Substitute the given values: $P = 360$ and $r = 0.03$.
- The equation representing the amount of money 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, assuming no deposits or withdrawals are made.
2. Recalling the Compound Interest Formula
The formula for compound interest is given by:
$$A = P(1 + r)^t$$
where:
- A is the final amount
- P is the principal amount (initial investment)
- r is the interest rate (as a decimal)
- t is the number of years
3. Identifying the Given Values
In this problem, we have:
- P = $360
- r = 3% = 0.03
- t = x (number of years)
- A = y (amount of money after x years)
4. Applying the Formula and Simplifying
Substituting these values into the compound interest formula, we get:
$$y = 360(1 + 0.03)^x$$
Simplifying the equation:
$$y = 360(1.03)^x$$
5. Final Answer
Therefore, the equation that represents the amount of money in Josiah's account after x years is:
$$y = 360(1.03)^x$$
### Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, you can use this formula to estimate how much your investments will grow over time. Similarly, when taking out a loan, understanding compound interest helps you calculate the total amount you'll need to repay. This knowledge empowers you to make sound choices about saving, investing, and borrowing money.
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