To determine the maximum amount Kylie can borrow given her constraints, we need to use the formula for the present value of an annuity. This formula is used to calculate the present value of a series of equal periodic payments. The formula is given by:
[tex]\[ P = \frac{PMT \left((1 + r)^n - 1\right)}{r (1 + r)^n} \][/tex]
Where:
- [tex]\( P \)[/tex] is the present value (the amount of the loan she can afford).
- [tex]\( PMT \)[/tex] is the monthly payment (1310 dollars).
- [tex]\( r \)[/tex] is the monthly interest rate.
- [tex]\( n \)[/tex] is the total number of payments.
Given the details:
- The APR is 8.4%, which we must convert to a monthly rate by dividing by 12:
[tex]\[ r = \frac{8.4\%}{12} = 0.084 \div 12 = 0.007 \][/tex]
- The duration of the loan is 25 years, which needs to be converted to months:
[tex]\[ n = 25 \times 12 = 300 \][/tex]
Substituting these values into the formula, we get:
[tex]\[ P = \frac{1310 \left((1 + 0.007)^{300} - 1\right)}{0.007 (1 + 0.007)^{300}} \][/tex]
Now, looking at the provided expressions, we match the correct formula representation:
A. [tex]\(\frac{(51310)\left((1+0.084)^{300}-1\right)}{(0.084)(1+0.084)^{300}}\)[/tex]
B. [tex]\(\frac{(51310)\left((1-0.084)^{300}-1\right)}{(0.084)(1+0.084)^{300}}\)[/tex]
C. [tex]\(\frac{(51310)\left((1+0.007)^{300}-1\right)}{(0.007)(1+0.007)^{300}}\)[/tex]
D. [tex]\(\frac{(51310)\left((1-0.007)^{300}-1\right)}{(0.007)(1+0.007)^{300}}\)[/tex]
The correct expression matches Option C:
[tex]\(\frac{(51310)\left((1+0.007)^{300}-1\right)}{(0.007)(1+0.007)^{300}}\)[/tex]
Therefore, the correct choice is:
C. [tex]\(\frac{(51310)\left((1+0.007)^{300}-1\right)}{(0.007)(1+0.007)^{300}}\)[/tex]
