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Answer :
Sure, let's break down the problem step-by-step.
Josiah invests \[tex]$360 into an account with an annual interest rate of 3%. We need to find the equation that represents the amount of money in his account, \( y \), after \( x \) years.
Here's how we derive the equation:
1. Initial Investment: Josiah starts with \$[/tex]360. This is our principal amount ([tex]\( P \)[/tex]).
2. Annual Interest Rate: The account earns 3% interest per year. As a decimal, this is 0.03.
3. Compound Interest Formula: The formula for compound interest, when the interest is compounded annually, is:
[tex]\[
A = P (1 + r)^n
\][/tex]
where [tex]\( A \)[/tex] is the amount after [tex]\( n \)[/tex] years, [tex]\( P \)[/tex] is the principal amount, [tex]\( r \)[/tex] is the annual interest rate, and [tex]\( n \)[/tex] is the number of years.
4. Substitute the Values:
- [tex]\( P = 360 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( n = x \)[/tex] (since we want the amount after [tex]\( x \)[/tex] years)
Plugging these values into the formula, we get:
[tex]\[
y = 360 (1 + 0.03)^x
\][/tex]
5. Simplify the expression inside the parentheses:
[tex]\[
y = 360 (1.03)^x
\][/tex]
Therefore, the equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years is:
[tex]\[
y = 360 (1.03)^x
\][/tex]
So, the correct answer is:
[tex]\[
y = 360 (1.03)^x
\][/tex]
Josiah invests \[tex]$360 into an account with an annual interest rate of 3%. We need to find the equation that represents the amount of money in his account, \( y \), after \( x \) years.
Here's how we derive the equation:
1. Initial Investment: Josiah starts with \$[/tex]360. This is our principal amount ([tex]\( P \)[/tex]).
2. Annual Interest Rate: The account earns 3% interest per year. As a decimal, this is 0.03.
3. Compound Interest Formula: The formula for compound interest, when the interest is compounded annually, is:
[tex]\[
A = P (1 + r)^n
\][/tex]
where [tex]\( A \)[/tex] is the amount after [tex]\( n \)[/tex] years, [tex]\( P \)[/tex] is the principal amount, [tex]\( r \)[/tex] is the annual interest rate, and [tex]\( n \)[/tex] is the number of years.
4. Substitute the Values:
- [tex]\( P = 360 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( n = x \)[/tex] (since we want the amount after [tex]\( x \)[/tex] years)
Plugging these values into the formula, we get:
[tex]\[
y = 360 (1 + 0.03)^x
\][/tex]
5. Simplify the expression inside the parentheses:
[tex]\[
y = 360 (1.03)^x
\][/tex]
Therefore, the equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years is:
[tex]\[
y = 360 (1.03)^x
\][/tex]
So, the correct answer is:
[tex]\[
y = 360 (1.03)^x
\][/tex]
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