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Answer :
To solve this problem, we need to determine which equation correctly represents the amount of money in Josiah's account after a certain number of years, given the initial investment and the interest rate.
1. Understand the Problem: Josiah invests $360 with an annual interest rate of 3%. He makes no additional deposits or withdrawals from the account. We need to find an equation that gives the amount of money, [tex]\( y \)[/tex], in the account after [tex]\( z \)[/tex] years.
2. Interest Type: The problem states that the interest is accrued annually at a fixed rate. This implies that we're dealing with compound interest.
3. Compound Interest Formula: The standard formula to calculate compound interest is:
[tex]\[
A = P(1 + r)^t
\][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
4. Plug in the Values:
- Principal [tex]\( P = 360 \)[/tex]
- Annual Interest Rate [tex]\( r = 3\% = 0.03 \)[/tex]
- We are solving for [tex]\( A = y \)[/tex] after [tex]\( t = z \)[/tex] years.
5. Formulate the Equation:
Substitute the values into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^z
\][/tex]
Simplifying this gives:
[tex]\[
y = 360(1.03)^z
\][/tex]
6. Select the Correct Equation: We identify this as the choice that matches one of the given options, specifically:
- [tex]\( y = 360(1.03)^z \)[/tex]
Thus, the correct equation that represents the amount of money in Josiah's account after [tex]\( z \)[/tex] years is:
[tex]\[
y = 360(1.03)^z
\][/tex]
1. Understand the Problem: Josiah invests $360 with an annual interest rate of 3%. He makes no additional deposits or withdrawals from the account. We need to find an equation that gives the amount of money, [tex]\( y \)[/tex], in the account after [tex]\( z \)[/tex] years.
2. Interest Type: The problem states that the interest is accrued annually at a fixed rate. This implies that we're dealing with compound interest.
3. Compound Interest Formula: The standard formula to calculate compound interest is:
[tex]\[
A = P(1 + r)^t
\][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
4. Plug in the Values:
- Principal [tex]\( P = 360 \)[/tex]
- Annual Interest Rate [tex]\( r = 3\% = 0.03 \)[/tex]
- We are solving for [tex]\( A = y \)[/tex] after [tex]\( t = z \)[/tex] years.
5. Formulate the Equation:
Substitute the values into the compound interest formula:
[tex]\[
y = 360(1 + 0.03)^z
\][/tex]
Simplifying this gives:
[tex]\[
y = 360(1.03)^z
\][/tex]
6. Select the Correct Equation: We identify this as the choice that matches one of the given options, specifically:
- [tex]\( y = 360(1.03)^z \)[/tex]
Thus, the correct equation that represents the amount of money in Josiah's account after [tex]\( z \)[/tex] years is:
[tex]\[
y = 360(1.03)^z
\][/tex]
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