We appreciate your visit to Josiah invests tex 360 tex into an account that accrues tex 3 tex interest annually Assuming no deposits or withdrawals are made which equation represents. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To determine which equation represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years, we need to understand how interest accrues over time.
### Step-by-Step Solution:
1. Identify the Initial Investment:
Josiah invests [tex]\(\$360\)[/tex].
2. Identify the Interest Rate:
The interest rate is [tex]\(3\%\)[/tex] annually.
3. Understand Annual Compounding:
Because the interest is compounded annually, each year the amount in the account grows by [tex]\(3\%\)[/tex].
4. Formula for Compound Interest:
The compound interest formula is:
[tex]\[
A = P(1 + r)^t
\][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money in the account after [tex]\( t \)[/tex] years,
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the number of years the money is invested.
5. Substitute the Given Values:
- [tex]\( P = 360 \)[/tex] (initial investment),
- [tex]\( r = 0.03 \)[/tex] (since [tex]\(3\% = \frac{3}{100} = 0.03\)[/tex]),
- [tex]\( t = x \)[/tex] (since we need the amount after [tex]\( x \)[/tex] years).
6. Formulate the Equation:
Substitute the values into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
y = 360(1.03)^x
\][/tex]
### Conclusion:
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]
This matches the option:
[tex]\[
y=360(1.03)^x
\][/tex]
So, the correct answer is: [tex]\( y=360(1.03)^x \)[/tex].
### Step-by-Step Solution:
1. Identify the Initial Investment:
Josiah invests [tex]\(\$360\)[/tex].
2. Identify the Interest Rate:
The interest rate is [tex]\(3\%\)[/tex] annually.
3. Understand Annual Compounding:
Because the interest is compounded annually, each year the amount in the account grows by [tex]\(3\%\)[/tex].
4. Formula for Compound Interest:
The compound interest formula is:
[tex]\[
A = P(1 + r)^t
\][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money in the account after [tex]\( t \)[/tex] years,
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the number of years the money is invested.
5. Substitute the Given Values:
- [tex]\( P = 360 \)[/tex] (initial investment),
- [tex]\( r = 0.03 \)[/tex] (since [tex]\(3\% = \frac{3}{100} = 0.03\)[/tex]),
- [tex]\( t = x \)[/tex] (since we need the amount after [tex]\( x \)[/tex] years).
6. Formulate the Equation:
Substitute the values into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
y = 360(1.03)^x
\][/tex]
### Conclusion:
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]
This matches the option:
[tex]\[
y=360(1.03)^x
\][/tex]
So, the correct answer is: [tex]\( y=360(1.03)^x \)[/tex].
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