An aeroplane leaves a point on latitude 54°S at 10 a.m. It flies due south at a steady speed of 800km\h.A what is its latitude at mid- day?B at what time does it pass over the south role? Unless told otherwise, use the take 40 000 for 2^R. Take log^ to be 0.4971.
A. Well, if the plane is flying due south, it means it will maintain the same longitude but change its latitude. Since it flies at a steady speed of 800 km/h and the mid-day is at 12 p.m, we can calculate the distance it would have traveled using the formula distance = speed × time.
So, in two hours, the plane would have traveled 800 km/h × 2 h = 1600 km.
Now, let's convert this distance into a change in latitude using the given conversion factor of 40,000 for 2^R. We can use the formula change in latitude = distance / conversion factor.
Change in latitude = 1600 km / 40,000 = 0.04.
Since the plane was initially at latitude 54°S, subtracting the change in latitude from the initial latitude gives us:
New latitude = 54°S - 0.04 = 53.96°S.
So, at mid-day, the latitude of the plane would be approximately 53.96°S.
B. Since the plane is flying due south, it will eventually cross the South Pole, which is located at latitude 90°S. We already calculated that the plane has a change in latitude of 0.04 from its initial position.
Therefore, to find the time it takes to cross the South Pole, we can use the formula: time = change in latitude / speed.
Time = 0.04 / 800 km/h = 0.00005 hours.
Converting this time to minutes, we get 0.00005 hours × 60 minutes/hour ≈ 0.003 minutes.
So, the plane will pass over the South Pole at approximately 12 p.m + 0.003 minutes, which is essentially still at mid-day.
Hope that adds a touch of humor to your flight-related question! Safe travels!
To determine the latitude of the airplane at mid-day, we need to calculate how far it has traveled in that time. Given that the airplane is flying due south at a steady speed of 800km/h, we can use the formula:
Distance = Speed x Time
Let's calculate the distance the airplane travels in 2 hours (from 10 a.m. to mid-day):
Distance = 800 km/h x 2 hours = 1600 km
To calculate the new latitude, we need to convert the distance traveled into an angle. We can use the following formula, which is derived from the relationship between distance and angle on a sphere:
Change in Latitude = (Distance / Circumference of Earth) x 360°
The circumference of the Earth (assuming a perfect sphere) is approximately 40,000 km. Substituting the given values:
Change in Latitude = (1600 km / 40000 km) x 360°
Change in Latitude = 0.04 x 360°
Change in Latitude = 14.4°
Since the airplane is flying due south from 54°S, we subtract the change in latitude from the initial latitude:
New Latitude = 54°S - 14.4°
New Latitude = 39.6°S
Therefore, the airplane's latitude at mid-day is 39.6°S.
To calculate at what time the airplane passes over the South Pole, we need to find out how long it takes to travel a distance equal to the circumference of the Earth. Substituting the given values:
Distance = 40,000 km
Speed = 800 km/h
Time = Distance / Speed
Time = 40,000 km / 800 km/h
Time = 50 hours
Since the airplane departs at 10 a.m., we add the travel time to this to determine the time it passes over the South Pole:
Passing Time = 10 a.m. + 50 hours
Passing Time = 12 p.m. (noon) on the following day
Therefore, the airplane passes over the South Pole at 12 p.m. (noon) on the following day.
To determine the latitude of the airplane at mid-day and the time it passes over the South Pole, we'll need to use some basic mathematical calculations.
A) To find the latitude at mid-day:
1. First, calculate the distance the airplane will have traveled by mid-day, using the fact that it travels at a steady speed of 800 km/h. Since it flies for 4 hours (from 10 a.m. to mid-day), the distance covered will be 800 km/h * 4 hours = 3200 km.
2. Next, we need to convert this distance into degrees of latitude. We know that 40,000 km represents a full circle around the Earth, and since the Earth is divided into 360 degrees of latitude, we can set up a proportion to find the corresponding number of degrees covered by 3200 km.
(3200 km) / (40,000 km) = (X degrees) / (360 degrees)
Solving for X, we get X = (3200 km * 360 degrees) / (40,000 km) = 288 degrees.
However, since the airplane is flying due south (latitude 54°S), we need to subtract this value from the starting latitude:
Latitude at mid-day = 54°S - 288° = -234°S.
Therefore, the latitude of the airplane at mid-day is 234°S.
B) To find the time the airplane passes over the South Pole:
1. The South Pole is located at 90°S latitude, so we need to determine the additional degrees the airplane has to cover to reach there from the starting latitude of 54°S.
Degrees to cover = 90°S - 54°S = 36°.
2. Using the same proportion as in part A above, we can find the distance the airplane needs to travel to cover the additional 36 degrees:
(Distance to cover) / 40,000 km = 36 degrees / 360 degrees
Solving for the distance to cover, we get:
Distance to cover = 40,000 km * (36 degrees / 360 degrees) = 4,000 km.
3. Since the airplane is flying at a steady speed of 800 km/h, we can calculate the time it will take to cover the remaining distance:
Time to cover = (Distance to cover) / (Speed) = 4,000 km / 800 km/h = 5 hours.
4. Adding this time to the initial departure time of 10 a.m., we determine when the airplane will pass over the South Pole:
Time = 10 a.m. + 5 hours = 3 p.m.
Therefore, the airplane will pass over the South Pole at 3 p.m.