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Answer :
- The problem involves calculating the future value of an investment with compound interest.
- The compound interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, and $t$ is the time in years.
- Substituting the given values, $P = 360$ and $r = 0.03$, into the formula gives $y = 360(1 + 0.03)^x$.
- Simplifying the equation results in the final answer: $\boxed{y = 360(1.03)^x}$.
### Explanation
1. Understanding the Problem
The problem describes a scenario of compound interest. We are given an initial investment, an annual interest rate, and the number of years. We need to find the equation that represents the amount of money in the account after a certain number of years.
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 annual interest rate (as a decimal)
- $t$ is the number of years
3. Identifying the Given Values
In this problem, we have:
- $P = 360$ (initial investment)
- $r = 0.03$ (3% annual interest rate as a decimal)
- $t = x$ (number of years)
- $A = y$ (amount of money after x years)
4. Substituting the Values into the Formula
Substituting these values into the compound interest formula, we get:
$$y = 360(1 + 0.03)^x$$
Simplifying the expression inside the parentheses:
$$y = 360(1.03)^x$$
5. Finding the Equation
Therefore, the equation that represents the amount of money in Josiah's account after $x$ years is:
$$y = 360(1.03)^x$$
This corresponds to the last option provided.
### Examples
Compound interest is a fundamental concept in finance. For example, if you invest money in a savings account or a certificate of deposit, the interest earned will compound over time, meaning you'll earn interest not only on your initial investment but also on the accumulated interest. Understanding compound interest can help you make informed decisions about saving and investing your money, as it allows you to project the potential growth of your investments over time. It's also applicable to loans, where interest accrues on the outstanding balance.
- The compound interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, and $t$ is the time in years.
- Substituting the given values, $P = 360$ and $r = 0.03$, into the formula gives $y = 360(1 + 0.03)^x$.
- Simplifying the equation results in the final answer: $\boxed{y = 360(1.03)^x}$.
### Explanation
1. Understanding the Problem
The problem describes a scenario of compound interest. We are given an initial investment, an annual interest rate, and the number of years. We need to find the equation that represents the amount of money in the account after a certain number of years.
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 annual interest rate (as a decimal)
- $t$ is the number of years
3. Identifying the Given Values
In this problem, we have:
- $P = 360$ (initial investment)
- $r = 0.03$ (3% annual interest rate as a decimal)
- $t = x$ (number of years)
- $A = y$ (amount of money after x years)
4. Substituting the Values into the Formula
Substituting these values into the compound interest formula, we get:
$$y = 360(1 + 0.03)^x$$
Simplifying the expression inside the parentheses:
$$y = 360(1.03)^x$$
5. Finding the Equation
Therefore, the equation that represents the amount of money in Josiah's account after $x$ years is:
$$y = 360(1.03)^x$$
This corresponds to the last option provided.
### Examples
Compound interest is a fundamental concept in finance. For example, if you invest money in a savings account or a certificate of deposit, the interest earned will compound over time, meaning you'll earn interest not only on your initial investment but also on the accumulated interest. Understanding compound interest can help you make informed decisions about saving and investing your money, as it allows you to project the potential growth of your investments over time. It's also applicable to loans, where interest accrues on the outstanding balance.
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