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 first need to calculate the length of the snake. We are given that one side of the snake forms an angle of 114° and is 220 m long. Using trigonometry, we can find the length of the snake's other side.
(a) Calculating the length of the snake's other side:
Let x be the length of the other side of the snake.
Using the sine function, we can write:
sin(114°) = x / 220
Rearranging the equation, we get:
x = 220 * sin(114°)
Using a calculator, we find that the length of the snake is approximately 316.79 m.
(b) Calculating the time difference between Inge and Olaf:
Both Inge and Olaf are running at a speed of 9.5 km/h, which is 9500 m/h. We can calculate the time it takes for each of them to reach the head of the snake.
For Inge, the time taken can be calculated by dividing the distance (316.79 m) by the speed (9500 m/h):
Time taken by Inge = 316.79 m / 9500 m/h
For Olaf, he needs to run along the snake, which has a total length of 420 m. The time taken by Olaf can be calculated by dividing the length of the snake (420 m) by the speed (9500 m/h):
Time taken by Olaf = 420 m / 9500 m/h
Now we can calculate the time difference between Inge and Olaf:
Time difference = Time taken by Inge - Time taken by Olaf
Substituting the values, we get:
Time difference = (316.79 m / 9500 m/h) - (420 m / 9500 m/h)
Simplifying the expression, we find that Inge reaches the head of the snake approximately 0.007 h (or 25.2 seconds) earlier than Olaf.
(c) To calculate the speed at which Olaf must run to reach the end of the snake at the same time as Inge, we can set up a proportion between their speeds and the distances they need to cover:
Inge's speed = 9.5 km/h
Olaf's speed = unknown (let's call it v km/h)
Distance covered by Inge = 316.79 m
Distance covered by Olaf = 420 m
The proportion can be set up as follows:
9.5 km/h / 316.79 m = v km/h / 420 m
Simplifying the proportion, we get:
9.5 km/h * 420 m = v km/h * 316.79 m
Now we can solve for v by dividing both sides of the equation:
v km/h = (9.5 km/h * 420 m) / 316.79 m
Using a calculator, we find that Olaf must run at a speed of approximately 12.58 km/h to reach the end of the snake at the same time as Inge.
To solve this problem, we can first calculate the time it takes for Inge to reach the head of the snake, and then calculate the time it takes for Olaf to reach the end of the snake while running along it.
(a) To find the time it takes for Inge to reach the head of the snake, we first need to find the distance she needs to cover. Using trigonometry, we can find the length of the snake along her path:
Length of snake along Inge's path = 220 m / cos(114°)
Next, we can calculate the time it takes for Inge to reach the head of the snake:
Time taken by Inge = Distance / Speed = Length of snake along Inge's path / 9.5 km/h
(b) To find the speed at which Olaf needs to run to reach the end of the snake at the same time as Inge, we first need to find the total distance he needs to cover. This is equal to the length of the snake:
Total distance for Olaf = 420 m
Next, we can calculate the time it takes for Olaf to reach the end of the snake:
Time taken by Olaf = Distance / Speed = Total distance for Olaf / Speed of Olaf
Since we want Inge and Olaf to finish the race at the same time, we can set their times equal:
Time taken by Inge = Time taken by Olaf
Now we can solve for the speed of Olaf:
Speed of Olaf = Total distance for Olaf / Time taken by Olaf
Keep in mind that we have already calculated the total distance for Olaf and the time taken by Inge in part (a).
By following these steps, you can find the solution to both parts of the problem.