A marathon runner runs at a steady rate of 15 km/h. When the runner is 7.5 km from the finish, a bird begins flying from the runner to the finish at 30 km/h. When the bird reaches the finish line it turns around and flies back to the runner, and then turns around again, repeating the back and forth trips until the runner reaches the finish line. How many kilometers does the bird travel?
How long does the bird fly in the air, or how long does the runner run? ans: t=distance/velocity=7.5km/15km/hr= 30 min
how far does the bird fly? velocity*time=30km/hr*1/2hr= 15km
15km
Well, let's see. If the bird is flying at 30 km/h and the runner is running at 15 km/h, it means the bird is twice as fast as the runner. So, every time the bird flies to the finish line and back, it covers a distance that is twice as long as the distance between the runner and the finish line.
Since the runner is initially 7.5 km away from the finish line, the bird will fly 7.5 km to the finish line and then 7.5 km back to the runner in the first roundtrip.
In the second roundtrip, the bird will fly 15 km to the finish line and then 15 km back to the runner.
In the third roundtrip, the bird will fly 30 km to the finish line and then 30 km back to the runner.
As you can see, the distances between the bird and the runner keep doubling with each roundtrip. So, the total distance the bird travels can be calculated by adding these distances:
7.5 + 15 + 30 + 60 + 120 + ...
We have ourselves a geometric series! The sum of an infinite geometric series can be calculated with the formula:
S = a / (1 - r),
where S is the sum, a is the first term, and r is the common ratio. In this case, a = 7.5 and r = 2.
Plugging in the values, we get:
S = 7.5 / (1 - 2) = 7.5 / (-1) = -7.5.
Uh-oh! It seems like the sum is negative. That means the infinite geometric series does not converge, and there is no finite total distance that the bird travels.
So, the bird basically travels an infinite number of kilometers. It's a real jet-setter!
To solve this problem, we need to determine how long it takes for the marathon runner to reach the finish line.
Distance = Rate x Time
The marathon runner runs at a steady rate of 15 km/h, and the remaining distance is 7.5 km. So, the time it takes for the runner to reach the finish line is:
Time = Distance / Rate
Time = 7.5 km / 15 km/h
Time = 0.5 hours
Now, we can calculate the distance the bird travels each time it goes back and forth between the runner and the finish line. The bird's speed is 30 km/h.
Distance = Rate x Time
Distance = 30 km/h x 0.5 hours
Distance = 15 km
Since there are multiple back and forth trips, we need to find the total distance the bird travels. Each round trip consists of two distances equal to 15 km, since the bird flies from the runner to the finish line and then back again.
Total distance traveled by the bird = 2 x Distance per round trip
Total distance traveled by the bird = 2 x 15 km
Total distance traveled by the bird = 30 km
Therefore, the bird travels a total of 30 kilometers.
To find out how many kilometers the bird travels, we first need to determine how long it takes for the runner to reach the finish line.
We can use the formula: time = distance / speed to find the time it takes for the runner to reach the finish line.
Given that the runner is 7.5 km away from the finish line and runs at a steady rate of 15 km/h:
time = 7.5 km / 15 km/h
time = 0.5 hours
Therefore, it takes the runner 0.5 hours to reach the finish line.
Now, let's find out how far the bird travels during this time.
The bird is flying at a constant speed of 30 km/h. Since both the runner and the bird are moving towards each other, the relative speed between them is the sum of their individual speeds.
Relative speed = 15 km/h + 30 km/h
Relative speed = 45 km/h
We can use the formula: distance = speed × time to find the distance covered by the bird in 0.5 hours.
distance = 45 km/h × 0.5 hours
distance = 22.5 km
Therefore, the bird travels 22.5 kilometers.