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You have just graduated from college and purchased a car for [tex]\$ 8000[/tex]. Your credit limit is [tex]\$ 12,000[/tex]. Assume that you make no payments, purchase nothing more, and there are no other fees. The monthly interest rate is [tex]14 \%[/tex]. Write an exponential equation to determine how much you will owe (represented by [tex]y[/tex]) after [tex]x[/tex] months with no more purchases or payments.

a. [tex]y=12,000(1.14)^x[/tex]
b. [tex]y=8,000(1.14)^x[/tex]
c. [tex]y=12,000(1.86)^x[/tex]
d. [tex]y=8,000(1.86)^x[/tex]

Please select the best answer from the choices provided:
A
B
C
D


Sagot :

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]
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