The biggest stuffed animal in the world is a snake 420 m long, constructed by Norwegian children. Suppose the snake is laid out in a park as shown in the figure below, forming two straight sides of a 114° angle, with one side 220 m long. Olaf and Inge run a race they invent. Inge runs directly from the tail of the snake to its head, and Olaf starts from the same place at the same time but runs along the snake.
(a) If both children run steadily at 9.5 km/h, Inge reaches the head of the snake how much earlier than Olaf?
(b) If Inge runs the race again at a constant speed of 9.5 km/h, at what constant speed must Olaf run to reach the end of the snake at the same time as Inge?
To solve this problem, we need to calculate the time it takes for both Inge and Olaf to reach their respective destinations.
(a) To find out how much earlier Inge reaches the head of the snake than Olaf, we first need to calculate the distances each of them have to cover.
Inge runs directly from the tail to the head of the snake, which is the hypotenuse of a right-angled triangle formed by the snake. Using the given information, we can use the Pythagorean theorem to find the length of this side. Let's call it h.
Using the Pythagorean theorem: a^2 + b^2 = c^2
Where a = 220m, b = ? (unknown side), and c = 420m.
Rearranging the equation, we get: b^2 = c^2 - a^2
Plugging in the values, we get: b^2 = 420^2 - 220^2 = 295,600 - 48,400 = 247,200
Taking the square root of both sides, we find b ≈ 497.17m
So, the distance Inge has to cover is approximately 497.17m.
Olaf, on the other hand, runs along the snake, which is a straight side of length 420m.
Now, we can calculate the time it takes for each of them to run their distances.
Time = Distance / Speed
For Inge:
Distance = 497.17m
Speed = 9.5 km/h = 9,500 m/60 min ≈ 158.33 m/min
Time taken by Inge = 497.17m / 158.33 m/min = 3.14 min (approximately)
For Olaf:
Distance = 420m
Speed = 9.5 km/h = 9,500 m/60 min ≈ 158.33 m/min
Time taken by Olaf = 420m / 158.33 m/min = 2.65 min (approximately)
Therefore, Inge reaches the head of the snake approximately 3.14 - 2.65 = 0.49 minutes, or about 29 seconds, earlier than Olaf.
(b) Now, let's calculate the speed at which Olaf must run to reach the end of the snake at the same time as Inge.
Inge takes 3.14 minutes to reach the head of the snake, so Olaf must also take 3.14 minutes to reach the end of the snake.
Distance covered by Olaf = Length of the snake - Distance covered by Inge
= 420m - 497.17m (approximately) = -77.17m (approximately)
Since distance cannot be negative, we need to ensure that Olaf stops running before the end of the snake. Therefore, there is no constant speed at which Olaf can run to reach the end of the snake at the same time as Inge.
In conclusion, Inge reaches the head of the snake approximately 29 seconds earlier than Olaf, and Olaf cannot reach the end of the snake at the same time as Inge.