A plane leaves an airport X, 20.6°E and 36.8°N, and flies due south along the same longitude
ong the for 8 hours at the rate of 1 000 km/h to another airport Y, 20.6°E and θS. The plane then flies west to another airport Z for 8 hours at the same speed. Calculate, to the nearest degree:
rotatio a the value of θ
b the longitude of Z.
(Take the radius of the Earth to be
int P
6400 k
To find the value of θS, we need to determine the distance traveled by the plane in the southward direction. We can use the formula:
distance = speed * time
In this case, the speed is 1000 km/h and the time is 8 hours:
distance = 1000 km/h * 8 hours = 8000 km
Since the plane is flying due south, the change in latitude from X to Y is -8000 km. We can divide this change in latitude by the circumference of the Earth to find the change in degrees:
change in θ = (change in latitude / circumference) * 360 degrees
The circumference of the Earth is given as 2 * π * 6400 km:
change in θ = (-8000 km / (2 * π * 6400 km)) * 360 degrees
change in θ = (-8000 / (2 * π * 6400)) * 360 degrees
Calculating this value will give us the change in θ, which we can add to the latitude of X to find the latitude of Y.
For part b, to find the longitude of Z, we need to determine the distance traveled by the plane in the westward direction. Since the plane is flying at the same speed of 1000 km/h for 8 hours, the distance traveled in the westward direction is:
distance = speed * time = 1000 km/h * 8 hours = 8000 km
To find the change in longitude from Y to Z, we can divide this distance by the circumference of a circle at the latitude of Y and multiply by 360 degrees:
change in longitude = (8000 km / (2 * π * radius of circle at latitude Y)) * 360 degrees
The radius of the circle at latitude Y can be found using the formula:
radius of circle at latitude Y = radius of Earth * cos(latitude of Y)
We are given the radius of Earth as 6400 km and the latitude of Y is θS (which we determined in part a). So:
radius of circle at latitude Y = 6400 km * cos(θS)
Finally, we can substitute the value of θS into the equation for the change in longitude to find the longitude of Z.
Let's calculate these values:
θS = (-8000 / (2 * π * 6400)) * 360
θS ≈ -45 degrees (rounded to the nearest degree)
radius of circle at latitude Y = 6400 km * cos(-45 degrees)
radius of circle at latitude Y ≈ 6400 km * 0.7071 ≈ 4530.67 km
change in longitude = (8000 km / (2 * π * 4530.67 km)) * 360 degrees
change in longitude ≈ 8000 / (2 * π * 4530.67) * 360 ≈ 35.34 degrees
longitude of Z = longitude of Y + change in longitude
longitude of Z = 20.6°E + 35.34° ≈ 55.94°E (rounded to the nearest degree)
So, the answers are:
a) θ ≈ -45 degrees
b) The longitude of Z is approximately 55.94°E.