1. ppr maths nbk
EXERCISE 2 (EARTH AS A SPHERE)
1. Given P, Q and R are three points on the surface of the earth. PQ is a diameter of
the parallel of latitude 20° N and PR is a diameter of the earth. The longitude of
R is 55° W.
(a) Find
(i) the longitude of P
(ii) the latitude of R (4 m)
(b) Find the distance between P and Q, in nautical miles, measured along the
common parallel of latitude. (4 m)
(c) An aeroplane took off from P at 2300 hours and flew towards Q via the
North Pole. If the aeroplane reached Q at 1500 hours on the next day, find
the average speed, in knots, of the aeroplane. (4 m)
2. K(60°S, 120°W), L, M and T are four points on the surface of the earth. KL is a
diameter of the earth.
(a) Find the distance between K and L, in nautical miles, measured along the
surface of the earth. (3 m)
(b) Find the latitude and longitude of L. (3 m)
(c) Given M is the midpoint of the distance KL, when measured via the South
Pole.Find the location of M. (3 m)
(d) Given T is 3 480 nautical miles due east of K. Find the longitude of T. (3 m)
3. Points X , Y and Z are on the surface of the earth along the parallel of latitude
56°S. The longitudes of X and Y are 40°W and 30°E respectively. YZ is a
diameter of the parallel of latitude 56°S.
(a) Find the longitude of Z. (2 m)
(b) Calculate the shortest distance between Y and Z, in nautical miles, measured
along the surface of the earth. (3 m)
(c) Calculate the distance between X and Y, in nautical miles, measured along
the parallel of latitude. (3 m)
(d) An aeroplane took of from Y and flew due north at a speed of 400 knots.
Find the time taken by the aeroplane to arrive at the North Pole. (4 m)

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4. J(50°N, 10°E), K(50°N, 50°W), L and M are four points on the surface of the
earth. JL is a diameter of the common parallel of latitude and M is due south of J.
(a) State the longitude of L. (2 m)
(b) Calculate
(i) the shortest distance between J and L, in nautical miles. (3 m)
(ii) the distance between J and K, in nautical miles, measured along the
common parallel of latitude. (3 m)
(c) An aeroplane took of from J and flew due south towards M at a speed of
650 knots. If the flight took 6 hours to reach M, find the latitude of M.
(4 m)
5. R(40°N, 80°W), S and T are three points on the surface of the earth. RS is the
diameter of a parallel of latitude 40°N. T is 3 600 nautical miles to the south of R.
(a) State the longitude of S. (2 m)
(b) Find the latitude of T. (3 m)
(c) Calculate the shortest distance, in nautical miles, from R to S measured
along the surface of the earth. (3 m)
(d) A ship sailed from S to R along the common parallel of latitude and then
due south to T. The total time taken for the journey was 20 hours.
Calculate the average speed of the ship for the whole journey. (4 m)
6. P(54°S, 75°E), Q and R are three points on the surface of the earth. PQ is the
diameter of parallel of latitude 54°S. R is 5 400 nautical miles to the north of P.
(a) Find the longitude of Q. (2 m)
(b) State the latitude of R. (2 m)
,
(c) Calculate the distance, in nautical miles, from P to Q measured along
the parallel of latitude. (4 m)
(d) An aeroplane took off from Q and flew towards P using the shortest
distance,as measured along the surface of the earth, and then flew due
north to R. Given that its average speed for the whole flight was 540
knots, calculate the total time taken for the flight. (4 m)

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7. A(0°, 40°W),B(0°, 50°E), C and D are four points on the surface of the
earth. D is on the north of A which is parallel of latitude 56°N.
(a) Calculate the distance, in nautical miles, from A to B measured along
the equator. (2 m)
(b) Calculate the shortest distance, in nautical miles, from C(56°N, 50°E) to
the North Pole. (3 m)
(c) An aeroplane took off from B and flew due north to C.Then it flew due
east to D and later due south to A.
Calculate
(i) the distance for the whole flight in nautical miles,
(ii) the total time taken for the whole flight if the average speed of the
plane is 160 knots. (7 m)
8. K(50°N, 55°E), L and M are three points on the surface of the earth. An
aeroplane took off from K and flew due east to L. KL is the diameter of
the parallel of latitude 50°N. Then the aeroplane flew back to K along the
shortest distance measured along the surface of the earth. Later the
aeroplane took off from K and flew due south to M which is 4 500
nautical miles from K.
(a) Find the longitude of L (2 m)
(b) Calculate the distance
(i) from K to L measured along the common parallel of latitude.
(ii) from L to K measured along the surface of the earth. (7 m)
(c) Find the latitude of M. (3 m)
9. P(67°N, 32°W), Q, R and T are four points on the surface of the earth. PQ is the
diameter of the parallel of latitude. R and T are at the equator where R is due
south to P and T is due south to Q.
(a) Calculate the shortest distance, in nautical miles, from P to R
measured along the surface of the earth. (2 m)
(b) Find the longitude of T if given that the shortest distance from R to
T as measured along the equator is 4 860 nautical miles. (4 m)

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(c) An aeroplane took off from P and flew due east to Q with an
average speed of speed of 600 knots.
Calculate
(i) the distance for the whole flight, in nautical miles,
(ii) the time taken for the whole flight. (6 m)
10. K ,L, M and P are four points on the surface of the earth. K and L lie on the
equator. M and P lie on the parallel of latitude 50°S.
(a) Given the shortest distance between K and L is 2 280 nautical miles. Find
the difference in longitude between K and L. (4 m)
(b) Given M and P are both due south of K and L respectively. Calculate the
distance between M and P, measured along the parallel of latitude. (4 m)
(c ) Find the distance between M and the South Pole. (4 m)
END OF QUESTION PAPER

5. ppr maths nbk
ANSWERS
1. a) i) 125° E 10. a) 38°
ii) 20° S b) 1 465.56 n.m.
b) 10148.68 n.m. c) 2 400 n.m.
c) 525 knots
2. a) 10 800 n.m.
b) 60° N, 60° E
c) M (60° S, 60° E)
d) 4° W
3. a) 150° W
b) 4 080 n.m.
c) 2 348.61 n.m.
d) 21 hours 54 minutes
4. a) 170° W
b) 4 800 n.m.
c) 2 314.03 n.m.
d) 15° S
5. a) 100° W
b) 20° S
c) 6 000 n.m.
d) 593.7 knots
6. a) 105° W
b) 36° N
c) 6 348.08 n.m.
d) 18 hours
7. a) 5 400 n.m.
b) 2 040 n.m.
c) 9 739.64 n.m.
d) 60.9 hours
8. a) 125° W
b) 6 942.11 n.m.
c) 4 800 n.m.
d) 25° S
9. a ) 4 020 n.m
b) 49° E
c) i) 4 219.9 n.m.
ii) 7 hours 2 minutes