two airplanes leave an airport at the same time one hour later they are 189km/hr apart. if one plane traveled 168km/hr and the other traveled 244km/hr for the hour what is the angle between their flight path
the answer is 50.6
Did you make your sketch?
You have a triangle with all sides known, a clear case of the cosine law,
let the angle between them be Ø
189^2 = 244^2 + 168^2 - 2(244)(168)cosØ
2(244)(168)cosØ = 244^2 + 168^2 - 189^2
cosØ = (244^2 + 168^2 - 189^2)/(2(244)(168) )
take over, let me know what you got, so I can check it
no
0.63?
Hmm, it seems like the pilots have started a game of "Who Can Fly the Farthest!" Well, let's find out the angle between their flight paths!
To do that, we can start by calculating how far each plane has traveled during that hour. The first plane, flying at 168 km/hr, will have covered a distance of 168 km in that hour. The second plane, zooming at 244 km/hr, will have traveled 244 km during the same time.
Now, we know that the two planes are 189 km apart after flying for one hour. To determine the angle between their flight paths, we can use some trigonometry! Specifically, we can use the cosine rule.
The cosine rule states that in a triangle with side lengths a, b, and c, and the angle opposite side length c is denoted as angle C, we can use the formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, let's assume the first plane's path is side a, the second plane's path is side b, and the distance between them is side c. We are interested in finding angle C.
Using the values we have, let's plug them into the formula:
(189)^2 = (168)^2 + (244)^2 - 2 * 168 * 244 * cos(C)
Cancelling out the common terms, we get:
35721 = 28224 + 59536 - 81408 * cos(C)
Now, let's simplify:
361 = -81408 * cos(C)
Dividing both sides by -81408, we obtain:
cos(C) = -361/81408
Using inverse cosine, we find:
C ≈ arccos(-361/81408)
At this point, I have to apologize, but my humorous database falls short of performing complex calculations like this. However, you can use a scientific calculator or an online calculator to determine the approximate value of angle C. Just remember to convert the measurement to degrees since most calculators use radians.
Safe and fun travels to both planes!
To find the angle between the flight paths of the two airplanes, we can use the concept of vectors. Let's break down the problem step by step.
Step 1: Determine the distance covered by each plane during the hour.
- One plane travels at a speed of 168 km/h for one hour, so it covers a distance of 168 km.
- The other plane travels at a speed of 244 km/h for one hour, so it covers a distance of 244 km.
Step 2: Determine the position vectors of each plane at the end of the hour.
- Let's assume the position of the slower plane at the end of the hour is represented by vector A, and the position of the faster plane is represented by vector B.
- Since the slower plane traveled 168 km in one hour, the magnitude of vector A is 168 km.
- Similarly, the magnitude of vector B is 244 km.
Step 3: Calculate the position vector between the two planes at the end of the hour.
- The position vector between the two planes is obtained by subtracting vector A from vector B, as it represents the displacement between the two planes.
- Let's denote this position vector as vector AB.
- The magnitude of vector AB can be determined using the distance formula, which is the square root of the sum of squares of the vector's components.
- In this case, the magnitude of vector AB is 189 km, as given in the question.
Step 4: Use the dot product formula to find the angle between the flight paths.
- The dot product of two vectors A and B can be calculated using the formula: A · B = |A| |B| cosθ, where θ is the angle between the vectors.
- Since we know the magnitudes of vectors A and B, as well as the dot product, we can rearrange the formula to solve for cosθ: cosθ = (A · B) / (|A| |B|).
- In this case, A · B is obtained by multiplying the magnitudes of vector A and B, and cosθ can be calculated by dividing the dot product by the product of the magnitudes.
Step 5: Calculate the angle between the flight paths.
- Use the inverse cosine function (commonly denoted as cos^(-1) or arccos) to find the angle, θ, based on the value of cosθ obtained in the previous step.
By following these steps, you should be able to calculate the angle between the flight paths of the two airplanes.