If Sandy can afford car payments of $$ \$ 300 $$ per month for 6 years, what is the price of a car that she can afford now? Assume an interest rate of 8.4 percent HW Score: $$ 49.2 \%, 33.95 $$ of 69 points Sandy can afford a car that costs $$ \$ \square $$ or less. (Round to the nearest dollar as needed.)

Question

UpStudy AI · Accepted Answer

Given:
- Monthly payment, $$ PMT = \$300 $$
- Loan term, $$ n = 6 \text{ years} \times 12 \text{ months/year} = 72 \text{ months} $$
- Annual interest rate, $$ 8.4\% $$; monthly rate, $$ r = \frac{0.084}{12} = 0.007 $$

We use the present value formula for an annuity (which represents the car price):

$$
PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}
$$

Substitute the given values:

$$
PV = 300 \times \frac{1 - (1 + 0.007)^{-72}}{0.007}
$$

Step 1. Calculate the term $$ (1 + 0.007)^{-72} $$:

$$
(1.007)^{-72} = \frac{1}{(1.007)^{72}}
$$

Step 2. Compute $$ (1.007)^{72} $$.  
Taking the natural logarithm:

$$
(1.007)^{72} = e^{72 \ln(1.007)}
$$

Approximating $$ \ln(1.007) \approx 0.006975 $$:

$$
72 \ln(1.007) \approx 72 \times 0.006975 = 0.5022
$$
$$
(1.007)^{72} \approx e^{0.5022} \approx 1.652
$$

Thus,

$$
(1.007)^{-72} \approx \frac{1}{1.652} \approx 0.6056
$$

Step 3. Substitute back into the PV formula:

$$
PV = 300 \times \frac{1 - 0.6056}{0.007}
$$
$$
PV = 300 \times \frac{0.3944}{0.007}
$$

Step 4. Compute the fraction:

$$
\frac{0.3944}{0.007} \approx 56.34
$$

Step 5. Finally, calculate the price:

$$
PV \approx 300 \times 56.34 \approx 16,902
$$

Rounding to the nearest dollar, Sandy can afford a car priced at approximately:

$$
\$16,902
$$
Sandy can afford a car priced at approximately \$16,902 or less.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Economics Questions