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Answer :
To determine which equation represents the amount of money in Josiah's account, we need to understand the compound interest formula. The formula for compound interest is given by:
[tex]\[ y = P (1 + r)^x \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( x \)[/tex] is the number of years the money is invested,
- [tex]\( y \)[/tex] is the amount of money in the account after [tex]\( x \)[/tex] years.
In this problem:
- Josiah invests [tex]\( P = 360 \)[/tex] dollars.
- The annual interest rate [tex]\( r = 0.03 \)[/tex] (3%).
Using the compound interest formula, we can set up the equation:
[tex]\[ y = 360 (1 + 0.03)^x \][/tex]
Simplifying inside the parentheses:
[tex]\[ y = 360 (1.03)^x \][/tex]
Thus, 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]
Let's confirm that this is the correct choice among the options provided:
1. [tex]\( y = 360 (1.3)^x \)[/tex]
2. [tex]\( y = 360 (0.3)^x \)[/tex]
3. [tex]\( y = 360 (0.03)^x \)[/tex]
4. [tex]\( y = 360 (1.03)^x \)[/tex]
After evaluating these options:
- [tex]\( y = 360 (1.3)^x \)[/tex] is incorrect because 1.3 is too large to represent a 3% interest rate.
- [tex]\( y = 360 (0.3)^x \)[/tex] is incorrect because 0.3 is too small and would indicate a substantial decrease, not an increase, in the principal.
- [tex]\( y = 360 (0.03)^x \)[/tex] is incorrect because 0.03 would represent a much smaller annual decrease, not the correct interest rate.
- [tex]\( y = 360 (1.03)^x \)[/tex] correctly represents the principal growing at a 3% annual interest rate.
Therefore, the correct equation is:
[tex]\[ y = 360 (1.03)^x \][/tex]
[tex]\[ y = P (1 + r)^x \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( x \)[/tex] is the number of years the money is invested,
- [tex]\( y \)[/tex] is the amount of money in the account after [tex]\( x \)[/tex] years.
In this problem:
- Josiah invests [tex]\( P = 360 \)[/tex] dollars.
- The annual interest rate [tex]\( r = 0.03 \)[/tex] (3%).
Using the compound interest formula, we can set up the equation:
[tex]\[ y = 360 (1 + 0.03)^x \][/tex]
Simplifying inside the parentheses:
[tex]\[ y = 360 (1.03)^x \][/tex]
Thus, 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]
Let's confirm that this is the correct choice among the options provided:
1. [tex]\( y = 360 (1.3)^x \)[/tex]
2. [tex]\( y = 360 (0.3)^x \)[/tex]
3. [tex]\( y = 360 (0.03)^x \)[/tex]
4. [tex]\( y = 360 (1.03)^x \)[/tex]
After evaluating these options:
- [tex]\( y = 360 (1.3)^x \)[/tex] is incorrect because 1.3 is too large to represent a 3% interest rate.
- [tex]\( y = 360 (0.3)^x \)[/tex] is incorrect because 0.3 is too small and would indicate a substantial decrease, not an increase, in the principal.
- [tex]\( y = 360 (0.03)^x \)[/tex] is incorrect because 0.03 would represent a much smaller annual decrease, not the correct interest rate.
- [tex]\( y = 360 (1.03)^x \)[/tex] correctly represents the principal growing at a 3% annual interest rate.
Therefore, the correct equation is:
[tex]\[ y = 360 (1.03)^x \][/tex]
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