An xyz-coordinate system is placed with its origin at the center of the earth, so that the equator is in the xy-plane, the North Pole has coordinates (0, 0, 3960), and the xz-plane contains the prime meridian. Find the coordinates (x, y, z) of Osaka, Japan, whose lotitude and longitude are 34.7 degrees north and 135.5 degrees east.
review spherical coordinates
x = r cos(long)cos(lat)
y = r sin(long)cos(lat)
z = r sin(lat)
Well, let me put on my GPS clown shoes and calculate the coordinates for Osaka, Japan!
First, let's take a look at the given information. We know that the North Pole has coordinates (0, 0, 3960) in this xyz-coordinate system with the origin at the center of the Earth.
Now, let's break down the coordinates of Osaka, Japan. The latitude is 34.7 degrees north, and the longitude is 135.5 degrees east.
Since the North Pole has a positive z-coordinate and is the most "up" you can go in this coordinate system, we can say that Osaka is south of the North Pole. Hence, the z-coordinate of Osaka will be negative.
As for the x-coordinate, we know that the xz-plane contains the prime meridian, which lies along the longitude 0 degrees. Since Osaka is located at 135.5 degrees east, it is on the opposite side of the prime meridian, so the x-coordinate will be negative.
So, let's calculate the coordinates.
The x-coordinate: -distance from the prime meridian, which is 180 degrees - 135.5 degrees = -44.5 degrees.
The y-coordinate: Since the xy-plane represents the equator, the latitude of Osaka doesn't affect the y-coordinate. Therefore, it will be 0.
The z-coordinate: -distance from the North Pole, which is 0 degrees - 34.7 degrees = -34.7 degrees.
Putting it all together, the coordinates of Osaka, Japan, in this coordinate system are approximately (-44.5, 0, -34.7).
To find the coordinates (x, y, z) of Osaka, Japan on the xyz-coordinate system, you'll need to use the latitude and longitude information provided.
1. Convert the latitude and longitude from degrees to radians:
- Latitude: 34.7 degrees = 34.7 * (π/180) radians ≈ 0.6068 radians
- Longitude: 135.5 degrees = 135.5 * (π/180) radians ≈ 2.3630 radians
2. Determine the distance from the center of the earth to the North Pole:
The coordinates of the North Pole are given as (0, 0, 3960) in the xyz-coordinate system. Therefore, the distance from the center of the earth to the North Pole (z-coordinate) is 3960.
3. Determine the x-coordinate:
The x-coordinate can be calculated using the formula: x = r * cos(θ) * cos(φ), where
- r is the distance from the center of the earth to a point (3960 in this case),
- θ is the longitude (2.3630 radians),
- φ is the latitude (0.6068 radians).
Substitute the values: x = 3960 * cos(2.3630) * cos(0.6068) ≈ 2080.7926
4. Determine the y-coordinate:
The y-coordinate can be calculated using the formula: y = r * sin(θ) * cos(φ), where
- r is the distance from the center of the earth to a point (3960 in this case),
- θ is the longitude (2.3630 radians),
- φ is the latitude (0.6068 radians).
Substitute the values: y = 3960 * sin(2.3630) * cos(0.6068) ≈ 1892.7827
5. The coordinates (x, y, z) of Osaka, Japan are approximately (2080.7926, 1892.7827, 3960).
To find the coordinates (x, y, z) of Osaka, Japan, we need to use spherical coordinates and convert them to Cartesian coordinates within the xyz-coordinate system.
1. Start by converting the latitude and longitude of Osaka into radians. Recall that there are π radians in 180 degrees. In this case:
Latitude of Osaka (φ) = 34.7 degrees = (34.7/180) * π radians
Longitude of Osaka (θ) = 135.5 degrees = (135.5/180) * π radians
2. The formula to convert spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z) is as follows:
x = r * cos(θ) * sin(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(φ)
3. Substitute the values of φ and θ into the equations above to calculate the Cartesian coordinates (x, y, z) for Osaka, Japan:
x = 3960 * cos(135.5/180 * π) * sin(34.7/180 * π)
y = 3960 * sin(135.5/180 * π) * sin(34.7/180 * π)
z = 3960 * cos(34.7/180 * π)
4. Use a calculator to evaluate these expressions:
x ≈ -2198.96
y ≈ 3447.48
z ≈ 2854.03
Therefore, the coordinates (x, y, z) of Osaka, Japan, within the xyz-coordinate system are approximately (-2198.96, 3447.48, 2854.03).