Step 1 :We know that the total distance is 399 miles and they met after 7 hours. So, the sum of their speeds is \(\frac{399}{7} = 57\) mph.
Step 2 :Let's denote the bicycle's speed as x mph. Then the car's speed is x + 34 mph.
Step 3 :We can set up the equation x + x + 34 = 57 and solve it for x.
Step 4 :The solution to the equation is \(\frac{23}{2}\) which is 11.5 mph. This is the speed of the bicycle.
Step 5 :The speed of the car is 11.5 mph + 34 mph = 45.5 mph.
Step 6 :Final Answer: The average rate of the bicycle is \(\boxed{11.5}\) mph and the average rate of the car is \(\boxed{45.5}\) mph.