An aeroplane flies due north from Ikeja airport for 500km.if their flies on a bearing of 060 for a further distance of 300km before overflying a road junction . calculate the distance of the aeroplane from Ikeja airport when it was directly above the junction (b).the bearing of the aeroplane from Ikeja airport at this instance
.
Make your sketch.
label the start I, the point of turning K, and the junction as J
In triangle KIJ, you have angle JKI = 120°, KI = 500, KJ = 300
using the cosine law,
IJ^2 = 500^2 + 300^2 - 2(500)(300)cos120°
IJ = .... , (careful with the cos120°, it is negative)
Once you have IJ, use the sine law to find angle I
All angles are measured CW from +y-axis,
a. d = 500kmn[0o] + 300km[60o],
X = 500*sin0 + 300*sin60 = 259.8 km,
Y = 500*Cos0 + 300*Cos60 = 650 km,
d = sqrt(X^2 + Y^2) =
b. Tan A = X/Y,
A = ?.
To calculate the distance of the aeroplane from Ikeja airport when it was directly above the junction, we can use the concept of vectors. We'll break down the problem into two components: the northward distance and the eastward distance.
1. Northward Distance:
The aeroplane flies north from Ikeja airport for 500 km, so the northward distance is 500 km.
2. Eastward Distance:
The aeroplane then flies on a bearing of 060 for a further distance of 300 km. To calculate the eastward distance, we need to find the component of this distance in the eastward direction, which can be determined using trigonometry.
Drawing a diagram, we can see that the 060 bearing forms a right-angled triangle with the eastward distance as the hypotenuse. The angle opposite the eastward distance is 90 degrees minus 60 degrees (due to the right angle and the 060 bearing), which gives us an angle of 30 degrees.
Using trigonometry (cosine function), we can calculate the eastward distance:
Eastward distance = 300 km * cosine(30 degrees)
Eastward distance = 300 km * 0.866 (approx.)
Eastward distance ≈ 259.8 km
3. Distance from Ikeja airport to the junction:
To calculate the distance of the aeroplane from Ikeja airport when it was directly above the junction, we can use the Pythagorean theorem.
Distance from Ikeja airport to the junction = √(Northward distance^2 + Eastward distance^2)
Distance from Ikeja airport to the junction = √(500 km^2 + 259.8 km^2)
Distance from Ikeja airport to the junction ≈ √(250,000 km^2 + 67,416.04 km^2)
Distance from Ikeja airport to the junction ≈ √317,416.04 km^2
Distance from Ikeja airport to the junction ≈ 563.9 km
4. Bearing of the aeroplane from Ikeja airport at this instance:
To determine the bearing, we need to find the angle between the northward direction and a line connecting Ikeja airport and the junction.
Using trigonometry (tangent function), we can calculate the angle:
Tan(θ) = Eastward distance / Northward distance
Tan(θ) = 259.8 km / 500 km
Tan(θ) ≈ 0.5196
θ ≈ arctan(0.5196)
Therefore, the bearing of the aeroplane from Ikeja airport at this instance is approximately the angle θ we just calculated. You can use a calculator to find the exact value if needed.
To calculate the distance of the aeroplane from Ikeja airport when it was directly above the junction, we can break down the problem into two components: the northward distance and the eastward distance.
1. Northward Distance:
The plane flies due north from Ikeja airport for 500 km. This means that the northward distance the plane has covered is 500 km.
2. Eastward Distance:
The plane then flies on a bearing of 060 for a further distance of 300 km. This bearing indicates an angle of 60 degrees measured clockwise from true north.
To find the eastward distance, we can use trigonometry. We know that the adjacent side of a right-angled triangle represents the eastward distance, and the hypotenuse represents the total distance flown (300 km).
Using cosine:
cos(60°) = adjacent/hypotenuse
Adjacent = cos(60°) * 300 km
Adjacent = 0.5 * 300 km
Adjacent = 150 km
Now, we have the northward distance of 500 km and the eastward distance of 150 km. To find the total distance from Ikeja airport to the junction, we can use the Pythagorean theorem.
Total Distance = sqrt((Northward Distance)^2 + (Eastward Distance)^2)
Total Distance = sqrt((500 km)^2 + (150 km)^2)
Total Distance = sqrt(250000 km^2 + 22500 km^2)
Total Distance = sqrt(272500 km^2)
Total Distance ≈ 522.36 km
Therefore, the distance of the aeroplane from Ikeja airport when it was directly above the junction is approximately 522.36 km.
Now, let's find the bearing of the aeroplane from Ikeja airport at this instance. We can use trigonometry again.
Using tangent:
tan(60°) = opposite/adjacent
opposite = tan(60°) * 150 km
opposite = sqrt(3) * 150 km
opposite ≈ 259.81 km
Now we know the opposite side (259.81 km) and the adjacent side (150 km). To find the angle, we use the inverse tangent (arctan) function.
Bearing = arctan(opposite/adjacent)
Bearing = arctan(259.81 km/150 km)
Bearing ≈ 60.94°
Therefore, the bearing of the aeroplane from Ikeja airport when it was directly above the junction is approximately 60.94° (measured clockwise from true north).