Two towns P and Q lie on the same parallel latitude such that P is due east of Q. When local time at Q is 9.50am , the local time at P is 3.12pm.
( a) Find the longitude difference between P and Q (2mks)
( b) Given that the longitude of P is 52°E , find the longitude of Q (2mks)
( c) A pilot took off from town Q and flew to town P along the parallel of latitude. The pilot took 3 and a quarter hrs travelling at an average speed of 860km per hr to reach P. Calculate to 1dp the latitude of town P and Q if they both lie in the northern hemisphere (3mks)
( d) 2 towns R and S are both on the equator and 3820 nautical miles apart . Town R is due east of town S. Find the local time at R when the local time at S is 2.20pm . ( Take R= 6370km and Pi = 22/7) (3mks)
(a,b) there are 24 hours in a day, and 360° in a circle, so each hour spans 15° degrees of longitude.
(c,d) let the radius of the earth at the equator be Z. Then the radius of a circle at latitude θ is Z cosθ.
not 100 percent about this but here goes.
9:50am-3:12pm=5:22min=5.36666hrs 24hr=360degrees
(5.36666x360)/24= 80.50degrees difference
(a) To find the longitude difference between P and Q, we need to convert the time difference between them into hours.
Local time at Q = 9.50am
Local time at P = 3.12pm
The time difference is 3 hours and 22 minutes.
Converting 22 minutes into hours:
22 minutes = 22/60 hours = 0.367 hours
Therefore, the total time difference between P and Q is 3 hours + 0.367 hours = 3.367 hours.
Since the Earth completes one full rotation in 24 hours, the longitude difference is equal to the time difference divided by 24 hours, multiplied by 360 degrees.
Longitude difference = (3.367 hours / 24 hours) * 360 degrees = 50.01 degrees.
Therefore, the longitude difference between P and Q is 50.01 degrees.
(b) Given that the longitude of P is 52°E, we can calculate the longitude of Q using the longitude difference.
Longitude of Q = Longitude of P - Longitude difference
Longitude of Q = 52°E - 50.01 degrees = 1.99 degrees East.
Therefore, the longitude of Q is 1.99°E.
(c) The pilot took 3 and a quarter hours to travel from Q to P. To find the latitude of P, we need to calculate the distance traveled by the pilot.
Distance = Speed * Time
Distance = 860 km/hr * 3.25 hrs = 2795 km
As the pilot traveled along the parallel of latitude, the distance traveled is equal to the circumference of the latitude circle. The circumference of a circle can be calculated using the formula:
Circumference = 2πr
Since we're dealing with latitude and not longitude, the radius (r) is equal to the Earth's radius (R), which is approximately 6370 km.
Circumference of latitude circle = 2 * Pi * R = 2 * 22/7 * 6370 km = 40140 km (approximately)
So, the latitude of P is given by:
Latitude of P = Distance traveled / Circumference of latitude circle
Latitude of P = 2795 km / 40140 km = 0.0696 (approximately)
Latitude of Q = Latitude of P
Therefore, the latitude of both town P and Q is approximately 0.0696.
(d) The distance between town R and S is given as 3820 nautical miles.
To convert nautical miles to kilometers, we use the conversion factor: 1 nautical mile = 1.852 km.
Therefore, the distance between R and S is 3820 nautical miles * 1.852 km/nautical mile = 7070.64 km.
Since R and S are on the equator, the distance between them is equal to the circumference of the Earth at the equator.
Circumference of the Earth at the equator = 2 * Pi * R
Circumference of the Earth at the equator = 2 * 22/7 * 6370 km = 40040 km.
To find the time difference between R and S, we can set up a proportion:
Time difference at R / Time difference at S = Distance at R / Distance at S
Let's assume the time difference at R is x.
x / 2.20pm = 7070.64 km / 40040 km
Converting 2.20pm to hours:
2.20pm = 14 hours + 0.33 hours = 14.33 hours
x / 14.33 hours = 7070.64 km / 40040 km
Cross-multiplying:
x = (14.33 hours * 7070.64 km) / 40040 km
x = 2.526 hours
Therefore, the time difference at R is approximately 2.526 hours.
To find the local time at R when the local time at S is 2.20pm, we subtract the time difference from 2.20pm.
Local time at R = 2.20pm - 2.526 hours
Local time at R = 11.674 hours
Converting 11.674 hours to standard time format:
11 hours + 0.674 hours = 11 hours and 40.44 minutes
Therefore, the local time at R is approximately 11:40am.
(a) To find the longitude difference between P and Q, we need to calculate the time difference between their local times.
Given:
Local time at Q = 9.50 am
Local time at P = 3.12 pm
Convert both times to a 24-hour format:
Local time at Q = 09:50
Local time at P = 15:12
To calculate the time difference, we subtract the time at Q from the time at P:
15:12 - 09:50 = 5 hours and 22 minutes
Since P is due east of Q, the time difference between them corresponds to the longitude difference. Therefore, the longitude difference between P and Q is 5 hours and 22 minutes.
(b) Given the longitude of P is 52°E, we can use the longitude difference calculated in part (a) to find the longitude of Q.
Since the longitude difference is 5 hours and 22 minutes, we convert this time to degrees. One hour of time difference corresponds to 15 degrees of longitude.
Conversion:
5 hours x 15 degrees/hour = 75 degrees
22 minutes x (15 degrees/60 minutes) = 5.5 degrees
Therefore, the longitude of Q is 52°E - 75° + 5.5° = -17.5°E.
(c) To find the latitude of towns P and Q, we need to consider the time it took the pilot to fly from Q to P and the average speed of the flight.
Given:
Time taken to fly from Q to P = 3 and a quarter hours = 3.25 hours
Average speed = 860 km/h
Since the pilot flew along the parallel of latitude, the distance traveled is equal to the speed multiplied by the time:
Distance = Speed x Time
Distance = 860 km/h x 3.25 hours
Distance = 2795 km
The distance traveled along the parallel of latitude is equal to the circumference of the Earth at that latitude multiplied by the latitude difference between P and Q.
Circumference of Earth at given latitude = 2πR cos(latitude)
Since towns P and Q are on the same parallel latitude, their latitudes are the same, let's denote it as 'x'.
Distance = (2πR cos(x))(52°E - -17.5°E)
2795 km = (2πR cos(x))(69.5°)
2795 km = (69.5°)(2πR cos(x))
Solving for cos(x):
cos(x) = (2795 km) / [(69.5°)(2πR)] = 0.0203
Taking the inverse cosine on both sides:
x = cos^(-1)(0.0203) ≈ 1.553 radians
To convert the latitude from radian to degrees, multiply by 180/π:
x = 1.553 x (180/π) ≈ 89.0°
Therefore, the latitude of towns P and Q, when both lie in the northern hemisphere, is approximately 89.0°N.
(d) To find the local time at R when the local time at S is 2.20 pm, we need to calculate the time difference between them.
Given:
Distance between R and S = 3820 nautical miles = 7048 km
Radius of the Earth, R = 6370 km
π = 22/7
First, convert the distance between R and S to the longitude difference:
Longitude difference = (Distance / Circumference of Earth) x 360°
Circumference of Earth = 2πR
Longitude difference = (7048 km / (2π x 6370 km)) x 360° ≈ 32.9°
Now, we need to find the time difference corresponding to the longitude difference.
One hour of time difference corresponds to 15 degrees of longitude.
Time difference = (Longitude difference / 15°) x 1 hour
Time difference = (32.9° / 15°) x 1 hour ≈ 2.19 hours
To find the local time at R, add the time difference to the local time at S:
2.20 pm + 2.19 hours
Converting 2.19 hours to minutes:
0.19 x 60 minutes = 11.4 minutes
Adding the minutes to the local time:
2.20 pm + 2 hours + 11 minutes = 4.31 pm
Therefore, the local time at R when the local time at S is 2.20 pm is approximately 4.31 pm.