To determine how much you'll owe after [tex]\( x \)[/tex] months, we need to set up an equation that accounts for the principal amount and the compounded interest rate.
Given:
- Principal amount (initial amount borrowed) [tex]\( P = \$8,000 \)[/tex]
- Monthly interest rate [tex]\( r = 0.14 \)[/tex] (or 14%)
- Number of months [tex]\( x \)[/tex]
The formula for compound interest is:
[tex]\[ y = P(1 + r)^x \][/tex]
Where:
- [tex]\( y \)[/tex] is the amount owed after [tex]\( x \)[/tex] months
- [tex]\( P \)[/tex] is the principal amount
- [tex]\( r \)[/tex] is the monthly interest rate
- [tex]\( x \)[/tex] is the number of months
Substituting the given values into the formula:
[tex]\[ y = 8,000(1 + 0.14)^x \][/tex]
[tex]\[ y = 8,000(1.14)^x \][/tex]
So the correct exponential equation to determine how much you will owe after [tex]\( x \)[/tex] months is:
[tex]\[ y = 8,000(1.14)^x \][/tex]
From the choices provided:
a. [tex]\( y = 12,000(1.14)^x \)[/tex]
c. [tex]\( y = 12,000(1.86)^x \)[/tex]
b. [tex]\( y = 8,000(1.14)^x \)[/tex]
d. [tex]\( y = 8,000(1.86)^x \)[/tex]
The correct choice is:
B. [tex]\( y = 8,000(1.14)^x \)[/tex]
