A runner is jogging at a steady 2.7 km/hr. When the runner is 7.7 km from the finish line, a bird begins flying from the runner to the finish line at 5.4 km/hr (2 times as fast as the runner). When the bird reaches the finish line, it turns around and flies back to the runner. How far does the bird travel? Answer in units of km.
b. After this first encounter, the bird then turns around and flies from the runner back to the finish line, turns around again and flies back to the runner. The bird repeats the back and forth trips until the runner reaches the finish line. How far does the bird travel from the beginning (including the distance traveled to the first encounter)? Answer in units of km.
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To find the distance the bird travels, we can calculate the distance it covers in each encounter with the runner.
a. In the first encounter, the bird flies from the runner to the finish line, covering a distance of 7.7 km.
b. The bird then turns around and flies back to the runner. Since the bird is now flying at the same speed as the runner (2.7 km/hr), the time taken for the bird to travel back to the runner would be the same time it took to fly from the runner to the finish line.
To calculate this time, we use the formula: time = distance / speed.
So, the time taken for the bird to fly from the runner to the finish line is: 7.7 km / 5.4 km/hr = 1.43 hours.
Since it took the bird 1.43 hours to fly from the runner to the finish line, it will also take the same amount of time for the bird to fly back to the runner. Therefore, the bird covers a distance of 2 x 7.7 km = 15.4 km in one complete round trip (from the runner to the finish line and back to the runner).
c. The bird continues repeating these round trips until the runner reaches the finish line. So, we need to determine the number of round trips the bird will make.
The total distance the runner needs to cover is 7.7 km. The runner's speed is 2.7 km/hr, so the time required for the runner to reach the finish line is: 7.7 km / 2.7 km/hr = 2.85 hours.
Since each round trip takes 1.43 hours, we divide the total time taken by the runner by the time taken for each round trip: 2.85 hours / 1.43 hours = 2.
Therefore, the bird will make 2 round trips before the runner reaches the finish line.
To find the total distance the bird travels, we multiply the distance covered in one round trip by the number of round trips: 15.4 km x 2 = 30.8 km.
So, the bird travels a total distance of 30.8 km from the beginning (including the distance traveled to the first encounter).
To find the distance the bird travels, we need to consider the time it takes for the bird to reach the finish line from the runner's starting point, as well as the distance it travels during the back and forth trips.
First, let's find the time it takes for the bird to reach the finish line from the runner's starting point. We can use the formula:
Time = Distance / Speed
The bird travels a distance of 7.7 km at a speed of 5.4 km/hr. Plugging these values into the formula, we get:
Time = 7.7 km / 5.4 km/hr = 1.43 hours
Now, let's calculate the distance the bird travels during the back and forth trips. Since the bird is flying at twice the speed of the runner, the distance it travels in each trip is twice the distance the runner covers in the same time.
The runner's speed is 2.7 km/hr, so in 1.43 hours, the runner covers a distance of:
Distance = Speed * Time
Distance = 2.7 km/hr * 1.43 hours = 3.861 km
Since the bird travels twice the distance of the runner in each back and forth trip, the bird's total distance during the back and forth trips is:
Total Distance = 2 * Distance
Total Distance = 2 * 3.861 km = 7.722 km
Finally, to find the total distance the bird travels from the beginning (including the distance traveled to the first encounter), we add the distance the bird traveled to reach the finish line to the total distance during the back and forth trips:
Total Distance = 7.7 km + 7.722 km = 15.422 km
Therefore, the bird travels a total distance of 15.422 km from the beginning.