chapter 9- earth as a sphere(12)
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Mathematics F5TRANSCRIPT
Chapter 9 Earth as A Sphere
H.S.HOE
1. Sketch a great circle that passes through the North pole and the South pole and the point A(not hidden) or B (hidden) in each of the following diagrams below.
(a)
(b)
(c)
(d)
2. In each of the diagrams below, O is the centre of the earth and NGS is the Greenwich Meridian. State the longitude of points P and Q.
(a) Longitude of P =
Longitude of Q =(b) Longitude of P =
Longitude of Q =(c) Longitude of P =
Longitude of Q =
(d) Longitude of P =
Longitude of Q =(e) Longitude of P =
Longitude of Q =(f) Longitude of P =
Longitude of Q =
3. Given that NGS is the Greenwich meridian, sketch and label the meridian whose longitude is given in each of the diagrams below.(a) 35E and 120W
(b) 110 W and 70E
(c) 65E and 30W
(d) 45W and 135E
(e) 62E and 98E
(f) 45W and 90W
4. Find the different between the two given longitudes for the following. sketch and label the meridian whose longitude is given in each of the diagrams below.
(a) 24 E and 80 E
(b) 34 W and 72
(c) 56 E and 15223E
(d) 3540 E and 6430 W
(e) 2530 E and 4350 E
(f) 5441 E and 7819 W
1. Sketch a circle parallel to the equator and passes through point H for each of the following.(a)
(b)
(c)
(d)
2. In the diagrams below, O is the centre of the earth. State the latitudes of points C and D.Latitude of P =
Latitude of Q =
Latitude of R =Latitude of P =
Latitude of Q =
Latitude of R =Latitude of P =
Latitude of Q =
Latitude of R =
Latitude of P =
Latitude of Q =
Latitude of R =Latitude of P =
Latitude of Q =
Latitude of R =Latitude of P =
Latitude of Q =
Latitude of R =
3. In each of the following diagrams sketch and label the altitude for each of the following point P and Q. (a) P(12 S) and Q(68 S)
(b) P(23 N) and Q(35 S)
(c) P(20 N) and Q(73 N)
(d) P(30 S) and Q(45 N)
(e) P(10 S) and Q(10N)
(f) P(25 N) and Q( 85N)
4. Find the difference between the latitude given below.(a) 45N and 55N
Difference =
(b) 16.3N and 48S(c) 17S and 31S(d) 5S and 28.5N
(e) 18.2N and 65.6S(f) 4020N and 75.2N(f) 1034S and 7856S(g) 35S and 2916N
1. Study the figure carefully and complete the table below.
2. Study the figure carefully and complete the table below.
3. Study the figure carefully and state the position of the point P, Q and RP = ( )
Q = ( )
R = ( )
4. Sketch and label the position for each of the following points.(a) P(35 N, 25 W) and
Q(50 S, 60 E)
(b) P(40 N, 3 E) and
Q(32 N, 46 W)
(c) P(28 N, 34 E) and
Q(30 S, 46 W)
(d) P(50 S, 34 E) and
Q(40 N, 75 W)
(e) P(82 N, 25 E) and
Q(20 N, 125 W)
(f) P(15 N, 145 E) and
Q(52 S, 65 W)
1. For each of the following, calculate the distance, in nautical miles, of ABC.
2. Find the distance along the great circle (along the same meridian)between each pair of points.(a) P(15 N, 56 W) and
Q(52 S, 56 W)
(b) P(14 S, 45 E) and
Q(70 S, 45 E) (c) P(20 S, 33 W) and
Q(52 N, 33 W)
(d) P(10 N, 115 E) and
Q(52 N, 115 E)
(e) P(15 N, 45 W) and Q is
located at the north pole.(f) P(25 N, 75 E) and Q is
located at the south pole.
3. Sketch and calculate the distance of PQ in each of the following measure along the equator.(a) P(0 , 36 E) and
Q(0 , 75 E)
(b) P(0 , 13 E) and
Q(0 , 105 E)
(c) P(0 , 46 E) and
Q(0 , 25 W)
(d) P(0 , 70 W) and
Q(0 , 20 W)
(e) P(0 , 57 W) and
Q(0 , 28 E)
(f) P(0 , 125 W) and
Q(0 , 40 E)
1. In each of the diagrams, find the value x and hence find the latitude of the point Q.(a) PQ = 5400 n.m.
(b) PQ =1230 n.m.
(c) PQ = 4800 n. m.
(d) PQ = 2450 n. m.
(e) PQ = 2800 n.m.
(f) PQ = 6150 n.m.
2. A and B are two point on the surface of the earth. Find the latitude of B.(a) B is located 1500 n.m. due south of A(40N , 20 E)(b) B is located 1200 n.m. due north of A(15N , 115 W)(c) B is located 2100 n.m. due south of A(22S , 50 W)
(d) B is located 3000 n.m. due south of A(30N , 65 E)(e) B is located 1500 n.m. due south of A(25N , 20 E)(f) B is located 1500 n.m. due south of A(13S , 15 W)
1. In each of the following diagrams, O is the centre of the earth. Given the distance PQ along the equator, calculate the longitude of the point Q.(a) Distance PQ = 2500 n.m.
(b) Distance PQ = 1560 n.m.
(c) Distance PQ = 3450 n.m.
(d) Distance PQ = 4000 n.m.
2. C and D are two points on the surface of the earth. Find the longitude of D.
(a) D is located 3000 n.m. due east of C(0 , 20 E)
(b) ) D is located 1500 n.m. due east of C(0 , 25 W)(c) D is located 4500 n.m. due west of C(0 , 40 W)
(d) D is located 2855 n.m. due west of C(0 , 173 W)
(e) D is located 4200 n.m. due west of C(0N , 51 E)(f) D is located 3000 n.m. due east of C(0N , 145 E)
1. Sketch and find the shortest distance between A and B, measured along a parallel of latitude.(a) A(10 N, 42 W) and B(10 N, 70 W)
(b) A(45 N, 10 E) and B(45 N, 70 E)
(c) A(30 N, 25 E) and B(30 N, 25 W)
(d) A(70 N, 45 E) and B(70 N, 20 W)
(e) A(50 S, 70 W) and B(50 S, 110 W)
(f) A(15 S, 80 E) and B(15 N, 142 E)
2. Given the distance P and Q, measured along a parallel of latitude, find the longitude of Q.
(a) Q is situated 1500 n.m. due east of P(60 N, 10 W)
Let = difference between the two longitude
=
=
=
Hence,
longitude of Q =
=
(b) Q is situated 2400 n.m.due east of P(30 N, 10 E)(c) Q is situated 3500 n.m. due west of P(40 S, 82 E)
(d) Q is situated 1240 n.m.due west of P(34 S, 54 W)
(e) Q is situated 2315 n.m. due west of P(58 N, 12 E)(f) Q is situated 2100 n.m. due east of P(70 S, 150 W)
1. Find the shortest distance between A and B.(a) A(40N, 10 W) and B(60N, 170E)
(b) A(30 N, 35 W) and B(45N, 145E)
(c) A(55 S, 124 W) and B(55 S, 56E)
(d) A(80 N, 150 W) and B(10 S, 30E)
(E) Finding the Shortest distance between two points along the equator or the great circle
(e) A(60 S, 110 E) and B(45 N, 110E)
(f) A(0, 56 W) and B(0, 112 E)
(1) A(0, 130 W) and B(0, 4530W) are two points on the surface of the earth. An aeroplane took 15 hours to fly from A to B along the equator.(a) Calculate the distance from A to B along the equator.
(b) Calculate the average speed of the aeroplane from A to B(2) K(20S, 36E) and L(20S, 144W) are two points on the surface of the earth. An aeroplane flew at a speed of 650 knots from K to L along the parallel of latitude.
(a) Find the distance travelled by the aeroplane.
(b) Find the time, in hours and minutes, taken by the aeroplane to fly from K to L.
(3) P(0, 45W), Q(0, 53W), R(35N, 80E) and S(35N, 53W) are four points on the suaface of the earth.
(4) A(50N, 15W), B(50N, 20E) and C are three points on the surface of the earth.
(1)
Point
Latitude
Longitude
P
Q
R
The table shows the latitudes of points P, Q and R on the surface of the earth.
(a) Find the value of x if PQ is the diameter of the
earth
(b) Find the value of y if QR is the diameter of the
parallel of latitude 50S(c) Hence,
(i). Calculate the shortest distance of PR and QR
along the meridian.
(ii) Calculate the distance of QR measured along the
common parallel of latitude 50S.
(2)
In the diagram, N is the North Pole, S is the South Pole and NOS is the axis of the axis of the earth. Given that PQ is the diameter of the earth, QR is the diameter of the parallel of latitude.
(a) State the positions of P, Q and R.
(b) Calculate the distance of PR, measured along the
same meridian.
(c) Calculate the distance of QR, measured along the
common parallel of latitude.
(3) P(30N, 50 E) and Q(30N, 20W) and R are
three points on the surface of the earth.
(a) Calculate the shortest distance, in nautical miles,
from P to the North Pole.
(b) Given that R is 3600 nautical miles due south of
Q. Calculate the latitude of R.
(c) An aeroplane took off from P and flew westwards
to Q along the common parallel of latitude. The
average speed of the flight is 550 knots. Calculate
the total time taken by the aeroplane for the
journey.
(4) P(65N, 125E), Q, R and T are four points of the
same great circle on the surface of the earth. PQ
is the diameter of a great circle. The shortest
distance of PR through North Pole is 3000 nautical
miles. The shortest distance of QT through South
Pole is 3600 nautical miles.
(a) State the positions of Q, R and T.
(b) Calculate the distance of PR measured along the
common parallel of latitude.
(c) An aeroplane took off from P and flew due south
to T. The total time taken by the aeroplane for the
flight is 12 hours, find the average speed of the
aeroplane.
5). P(60S, 70E),Q and R are three points on the
surface of the earth. PQ is the diameter of the
earth and R is 5400 nautical miles due east of Q.
(a) State the positions of Q and R.(b) Calculate the shortest distance, in nautical miles, from
Q to R.
(c) An aeroplane took off from Q and flew towards R
using the shortest distance, and then flew due south to
P. The total time taken for the whole flight is 16 hours,
find the average speed of the aeroplane.
6). A(35S, 40E), B(35S, 20W),and C are three
points on the surface of the earth.
(a) Calculate the shortest distance, in nautical miles,
from A to South Pole measured along the surface
of the earth.
(b) Given that AC is the diameter of the earth. State
the position of C.
(c) An aeroplane took off from A and flew westwards
to B along the common parallel of latitude. The
average speed of the flight is 450 knots. Calculate
the time taken by the flight.
9.1 Longitud
1. A Great circle circle on the surface of the earth which has a diameter that passes the centre of the earth.
2. A Meridian is half of the great circle that connects the North pole and the South pole.
3. The Greenwich Meridian has a longitude of 0, and is chosen as reference circle for measuring longitude.
4. The longitude of a meridian is the angle to the east or west of the Greenwich Meridian. (x E) or (y W)
Chapter 9 Earth as A Sphere
9.2 Latitude
1. The latitude is stated as an angle to the north or south of the equator.
2. The parallel of latitude is a circle on the surface of the earth that is parallel to the equator.
3. All point on the same parallel latitude have the same latitude.
4. The north pole has latitude 90N and the south pole 90S.
45 E
125 E
Greenwish Meridian, 0
55 W
9.2 Latitude Finding the difference between the two latitudes
1. If the longitudes of A and B are on the same side of the Greenwich meridian xE and y E and(y > x)
Then the different between them = y - x
2. If the longitudes of A and B are on the opposite side of the Greenwich meridian are x E and y W
Then the different between them = x + y
(1).If the latitudes of A(x N)
and B(y N) or A(x S) and
B(y S) are on the same side
of the equator, so the angle
between them is the difference of their latitudes.
Thus,
Difference = y - x (y > x)
Example 1.
Difference = 79 - 45 = 34
Example 2.
Difference = 85 - 32 = 53
(1).If the latitudes of A(x N) and B(y S) or A(x S) and B(y N) are on the opposite side of the equator, so the angle between them is the sum of their latitudes.
Thus, Difference = y + x
Example:
Difference = 85 + 50 = 135
1
2
9.3 Location of a Place
1. The location of a place on the surface of the earth is determined by its latitude and longitude.
2. The location of a place is written as ( its latitude, its longitude) = ( x N/S, y E/W)
Example : A point P has a latitude of 25 N and 80 E
Then its location is written as P(25 N, 80 E)
9.4 Distance on the Surface of the Earth
(a)
PointLatitudeLongitudeLocationPQRS
(c)
(b)
PointLatitudeLongitudeLocationABCPQR
(A) Finding the distance between two points along a great circle.
Distance between A and B = EMBED Equation.3 the angle substance by the arc at the centre of the earth ( EMBED Equation.3 )
= EMBED Equation.3
The distance AB = 60 x
= 60 x ( EMBED Equation.3 ) nautical miles
The distance AB = 60 x
= 60 x EMBED Equation.3 nautical miles
= EMBED Equation.3
= EMBED Equation.3
= EMBED Equation.3
= EMBED Equation.3
9.4 Distance on the Surface of the Earth
(B) Finding the latitude of point B, given the latitude of point A and the distance AB along the same meridian
EMBED Equation.3
Distance measure along equator
Distance measure along meridian
9.4 Distance on the Surface of the Earth
(C) Finding the latitude of point Q, given the longitude of point P and the distance PQ along the equator.
EMBED Equation.3
9.4 Distance on the Surface of the Earth
(C) Finding the distance between two points along a parallel of latitude. All the parallels of latitudes other than equator are called smaller circles. The radius of the smaller circle (r) must be smaller than the radius of the earth or the great circles.
From EMBED Equation.3
Cos EMBED Equation.3
and
EMBED Equation.3
The difference in longitude
= EMBED Equation.3
= 35 + 24
= 59
Hence,
The distance PQ
= EMBED Equation.3
= 59 x 60 x cos60
= 1770 n.m.
The difference in longitude,
EMBED Equation.3
The distance AB measured along latitude
=
=
Hence, the shortest distance
BA = BNA
= 60 x 90
= 5400n.m.
EMBED Equation.3 BOA =180- 70-20
= 90
From EMBED Equation.3
Cos EMBED Equation.3
and
EMBED Equation.3
9.4 Distance on the Surface of the Earth
(D) Finding the Shortest distance between two points across North and South Pole
EMBED Equation.3 BOA =180- 80-35
= 65
Hence, the shortest distance
BA = BSA
= 60 x 65
= 3900n.m.
2
1
Problem Solving
Problem Solving SPM Format
(a) Mark the location of the points P, Q, R and T on the diagram.
(b) Find the distance, in nautical miles, from P to Q along the equator.
(c) Find the distance, in nautical miles, from R to T along the parallel of
latitude.
(d) An aeroplane took off from Q and flew due north along the longitude 53W
to T with average speed of 700 knots. Find the time taken by the plane.
Different
=
=
Different
=
=
(a) Calculate the shortest distance, in nautical miles from A to the North pole.
(b) Given that C is situated 3660 n.m. to the south of B, find the latitude of C.
(c) Mark A, B and C on the diagram given.
(d) An aeroplane flew east from A to B at a speed of 600 konts. If the aeroplane
arrived at B at 1115 hours, find the departure time of the aeroplane from A.
Different
=
=
Different
=
=
Different
=
=
Different
=
=
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