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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

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