Step 1 :First, calculate the total cost of the car including sales tax and license and title fees. The total cost is the purchase price minus the trade-in value, plus the sales tax on the reduced price, plus the license and title fees. So, the total cost is \(\$30,400 - \$13,200\) times \(1 + 0.075\) plus \$350, which equals \$18840.
Step 2 :Next, calculate the amount financed. This is the total cost of the car minus any down payment. In this case, there is no down payment, so the amount financed is also \$18840.
Step 3 :Finally, calculate the total installment price. This is the monthly payment times the number of payments. So, the total installment price is \$499.68 times 72, which equals \$35977.76.
Step 4 :Thus, the total installment price is \(\boxed{\$35977.76}\).