Final answer:
Joshua can afford a car loan payment of $230 per month after deducting other vehicle-related expenses from his total budget of $290. Using the loan formula, we can calculate the maximum loan amount he can take out, factoring in the 10% annual interest rate, a 6-year loan term, and the $2,000 down payment from his trade-in.
Step-by-step explanation:
Joshua is looking to purchase a new car and needs to determine the most he can afford to pay. The first step is to calculate his monthly budget for the car loan payment. Joshua can afford to pay $290 per month for the car, but we need to account for $60 per month in taxes, insurance, fees, and maintenance. Thus, the amount available for the car payment itself is $290 - $60 = $230 per month.
Next, we use the loan formula to find out the maximum loan amount Joshua can take out, considering the $2,000 from the trade-in as a down payment. The formula for the monthly payment on an installment loan is:
M = P [i(1+i)^n] / [(1+i)^n - 1]
Where:
M = monthly payment
I = monthly interest rate (annual rate/12)
N = number of payments (years*12)
P = principal amount (loan amount)
With a 10% annual interest rate, the monthly rate is 0.10 / 12. The number of payments over 6 years is 6 * 12. Therefore, P can be found by rearranging the loan formula to solve for the principal:
P = M [(1+i)^n - 1] / [i(1+i)^n]
Substituting the values we have:
P = $230 x [(1 + 0.10/12)^(6*12) - 1] / [(0.10/12) * (1 + 0.10/12)^(6*12)]
Calculating this gives the maximum loan amount Joshua can afford. Finally, to find the total price of the car he can afford, we add the $2,000 down payment to the loan amount calculated.