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Answer :
To determine the correct equation for the amount of money in Josiah's account after [tex]$x$[/tex] years with an annual interest rate of 3%, you need to understand how interest compounds.
Here's a step-by-step breakdown:
1. Identify the Principal: This is the initial amount of money, which in Josiah's case is [tex]$360.
2. Determine the Interest Rate: The interest rate given is 3%. In decimal form, this is 0.03.
3. Compound Interest Formula: The formula used to calculate the compound interest over time is:
\[
y = \text{Principal} \times (1 + \text{Interest Rate})^x
\]
- Here, \(y\) represents the amount of money in the account after \(x\) years.
- \(\text{Principal}\) is $[/tex]360.
- [tex]\(\text{Interest Rate}\)[/tex] is 0.03.
- [tex]\(x\)[/tex] is the number of years the money is invested.
4. Substitute the Known Values into the Formula:
- The formula becomes:
[tex]\[
y = 360 \times (1 + 0.03)^x
\][/tex]
- Simplifying inside the parentheses gives:
[tex]\[
y = 360 \times (1.03)^x
\][/tex]
5. Match the Equation with the Given Options:
- Among the options provided, the equation [tex]\(y = 360(1.03)^x\)[/tex] matches the one we derived.
So, the correct equation that represents the amount of money in Josiah's account after [tex]\(x\)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
Here's a step-by-step breakdown:
1. Identify the Principal: This is the initial amount of money, which in Josiah's case is [tex]$360.
2. Determine the Interest Rate: The interest rate given is 3%. In decimal form, this is 0.03.
3. Compound Interest Formula: The formula used to calculate the compound interest over time is:
\[
y = \text{Principal} \times (1 + \text{Interest Rate})^x
\]
- Here, \(y\) represents the amount of money in the account after \(x\) years.
- \(\text{Principal}\) is $[/tex]360.
- [tex]\(\text{Interest Rate}\)[/tex] is 0.03.
- [tex]\(x\)[/tex] is the number of years the money is invested.
4. Substitute the Known Values into the Formula:
- The formula becomes:
[tex]\[
y = 360 \times (1 + 0.03)^x
\][/tex]
- Simplifying inside the parentheses gives:
[tex]\[
y = 360 \times (1.03)^x
\][/tex]
5. Match the Equation with the Given Options:
- Among the options provided, the equation [tex]\(y = 360(1.03)^x\)[/tex] matches the one we derived.
So, the correct equation that represents the amount of money in Josiah's account after [tex]\(x\)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
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