Two planes leave the same airport at the same time. one flies 20 degrees east of north at 500 mph. the second flies 30 degrees east of south at 600 mph. how far apart are the planes after 2 hours?
I have a triangle with sides 1000 and 1200 with an angle of 130° between them.
Clearly a cosine law problem
d^2 = 1000^2 + 1200^2 - 2(1000)(1200) cos 130°
= ...
d = √...
=
let me know what you get.
Well, looks like they went on quite the adventure! Let's calculate their distance after 2 hours.
The first plane flies 20 degrees east of north, so it's going towards the northeast direction. Since it's flying at 500 mph, after 2 hours it will have traveled 1000 miles.
The second plane flies 30 degrees east of south, so it's heading towards the southeast direction. At 600 mph, it would have flown 1200 miles after 2 hours.
Now, let's imagine a triangular adventure that they embarked upon! The distance between the planes forms the hypotenuse of a right triangle created by their paths.
Using some trigonometry magic, we can find that this triangular distance is approximately 1490 miles.
So, after 2 hours, these planes found themselves about 1490 miles apart. They must have been having a wild time up there!
To find the distance between the two planes after 2 hours, we can use their velocities and the concept of relative velocity.
First, let's break down the velocities of the planes into their north-south and east-west components.
For the first plane:
- Velocity: 500 mph
- Direction: 20 degrees east of north
- North-South Component: 500 * cos(20)
- East-West Component: 500 * sin(20)
For the second plane:
- Velocity: 600 mph
- Direction: 30 degrees east of south
- North-South Component: 600 * sin(30)
- East-West Component: 600 * cos(30)
Now, let's calculate the displacement of each plane after 2 hours by multiplying their respective components by the time (2 hours).
For the first plane:
- North-South Displacement: (500 * cos(20)) * 2
- East-West Displacement: (500 * sin(20)) * 2
For the second plane:
- North-South Displacement: (600 * sin(30)) * 2
- East-West Displacement: (600 * cos(30)) * 2
To find the total displacement between the two planes, we need to sum the differences in their north-south and east-west displacements.
Total North-South Displacement: (500 * cos(20)) * 2 - (600 * sin(30)) * 2
Total East-West Displacement: (500 * sin(20)) * 2 + (600 * cos(30)) * 2
Now, we can use the Pythagorean theorem to find the overall distance between the two planes.
Distance = sqrt((Total North-South Displacement)^2 + (Total East-West Displacement)^2)
Plug in the values and calculate to find the distance.
To find the distance between the two planes after 2 hours, we can break it down into two steps:
Step 1: Find the displacement of each plane after 2 hours.
Step 2: Calculate the distance between the two displacements.
Step 1: Finding the displacement of each plane after 2 hours:
For the first plane flying 20 degrees east of north at 500 mph, we need to find the northward and eastward components of its velocity.
The northward component can be calculated using trigonometry:
northward component = velocity * cos(angle)
northward component = 500 mph * cos(20)
The eastward component can also be calculated using trigonometry:
eastward component = velocity * sin(angle)
eastward component = 500 mph * sin(20)
For the second plane flying 30 degrees east of south at 600 mph, we need to find the southward and eastward components of its velocity.
The southward component can be calculated using trigonometry:
southward component = velocity * cos(angle)
southward component = 600 mph * cos(30)
The eastward component can also be calculated using trigonometry:
eastward component = velocity * sin(angle)
eastward component = 600 mph * sin(30)
Step 2: Calculating the distance between the two displacements:
To calculate the distance between the two displacements, we will use the Pythagorean theorem:
distance = sqrt((northward displacement + southward displacement)^2 + (eastward displacement + eastward displacement)^2)
Once we have the values for the northward, southward, eastward, and eastward displacements, we can substitute them into the formula and calculate the distance between the two planes.
SO this problem is a cosine law problem?
And if i did the problem right...
It should be 7276.412812...
I changed the degree (130 degrees) to radians
If i should not change the degree to radians... it should be 1822.498064...