Suppose that a cyclist began a 399mi ride across a state at the western edge of the state, at the same time that a car traveling toward it leaves the eastern end of the state. If the bicycle and car met after $7 \mathrm{hr}$ and the car traveled $34 \mathrm{mph}$ faster than the bicycle, find the average rate of each.
The car's average rate is
(Type an integer or a decimal.)
Answer
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.
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.
