Earl wants to purchase a car. He secures an auto loan at 3.39% annual interest. His budget allows him to pay about $300 a month over the next 4 years. What price auto can he afford?

To solve for the price of the car that Earl can afford, we need to determine the present value of an annuity given the monthly payment, interest rate, and number of months he plans to make payments.

1. Identify the given information:
R = 300 \text{ (monthly payment)}
r = 0.0339 \text{ (annual interest rate)}
n = 4 \text{ (years)}
t = 4 \times 12 = 48 \text{ (months)}

2. Convert the annual interest rate to a monthly interest rate:
i = \frac{0.0339}{12} \approx 0.002825 \text{ (monthly interest rate)}

3. Use the formula for the present value of an annuity:
P = R \times \left( \frac{1 - (1 + i)^{-t}}{i} \right)

4. Plug in the values:
P = 300 \times \left( \frac{1 - (1 + 0.002825)^{-48}}{0.002825} \right)

5. Calculate the expression inside the parentheses:
P = 300 \times \left( \frac{1 - (1.002825)^{-48}}{0.002825} \right)
P = 300 \times \left( \frac{1 - \frac{1}{1.144773}}{0.002825} \right)
P = 300 \times \left( \frac{1 - 0.87355}{0.002825} \right)
P = 300 \times \left( \frac{0.12645}{0.002825} \right)
P \approx 300 \times 44.2229
P \approx 13,266.87

Therefore, the price of the car that Earl can afford is approximately:

P \approx \$13,266.87