Suppose that a cyclist began a 390 mi 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 meet after 6.5 hr and the car traveled 32.4 mph faster than the bicycle, find the average rate of each.
since distance = speed * time
6.5x + 6.5(x+32.4) = 390
To find the average rate of each, we need to determine the individual speeds of the cyclist and the car. Let's assume that the cyclist's speed is "x" mph.
Since the car travels 32.4 mph faster than the cyclist, we can express its speed as "x + 32.4" mph.
Now, let's calculate the distance traveled by each:
Distance traveled by the cyclist = Speed × Time = x mph × 6.5 hr
Distance traveled by the car = Speed × Time = (x + 32.4) mph × 6.5 hr
According to the problem, the total distance traveled by both the cyclist and the car is 390 miles. So we have the equation:
Distance traveled by the cyclist + Distance traveled by the car = 390 miles
(x mph × 6.5 hr) + ((x + 32.4) mph × 6.5 hr) = 390 miles
Now, let's solve this equation to find the value of x, which represents the cyclist's speed:
6.5x + 6.5(x + 32.4) = 390
6.5x + 6.5x + 210.6 = 390
13x + 210.6 = 390
13x = 390 - 210.6
13x = 179.4
x = 179.4 / 13
x ≈ 13.80
So, the cyclist's average speed is approximately 13.80 mph.
To find the average speed of the car, we can substitute this value back into the expression for the car's speed:
Car's speed = Cyclist's speed + 32.4
Car's speed ≈ 13.80 mph + 32.4 mph
Car's speed ≈ 46.20 mph
Therefore, the average rate of the cyclist is approximately 13.80 mph, while the average rate of the car is approximately 46.20 mph.