Question
A. An aeroplane flies from Town A (20°N,60°E) to town B(20°N,20°E). If the journey takes six hours, calculate correct to three significant figures, the average speed of the aeroplane.
B. If it then flies due north from town B to town C away, calculate correct to the nearest degree, the latitude of town c
B. If it then flies due north from town B to town C away, calculate correct to the nearest degree, the latitude of town c
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GPT-4o mini
To solve this problem, we will break it down into parts A and B.
### A. Calculating the average speed of the aeroplane
1. **Determine the distance between Town A and Town B.**
The coordinates of Town A are (20°N, 60°E) and Town B are (20°N, 20°E). Both towns are situated at the same latitude (20°N), so the distance between them can be calculated as the difference in longitude.
\[
\text{Longitude difference} = 60°E - 20°E = 40°
\]
To convert this angular distance to kilometers, we can use the formula for the distance along a circle of latitude:
\[
\text{Distance} = \text{(Longitude difference in degrees)} \times \left( \frac{ \pi \times R \times \cos(\text{latitude in radians})}{180} \right)
\]
where \( R \) is the average radius of the Earth (approximately 6371 km), and latitude in radians is \( 20° \times \frac{\pi}{180} \).
\[
\text{Distance} = 40° \times \left( \frac{ \pi \times 6371 \times \cos(20° \times \frac{\pi}{180})}{180} \right)
\]
Calculating \( \cos(20°) \):
\[
\cos(20°) \approx 0.9397
\]
Substituting values:
\[
\text{Distance} \approx 40 \times \frac{ \pi \times 6371 \times 0.9397}{180}
\]
\[
\text{Distance} \approx 40 \times \frac{20021.3 \times 0.9397}{180} \approx 40 \times 109.835 \approx 4393.4 \, \text{km}
\]
2. **Calculate average speed.**
The journey takes 6 hours, so the average speed \( v \) can be calculated as:
\[
v = \frac{\text{Distance}}{\text{Time}} = \frac{4393.4 \, \text{km}}{6 \, \text{hours}} \approx 732.2 \, \text{km/h}
\]
Correct to three significant figures, the average speed is:
\[
\boxed{732 \, \text{km/h}}
\]
### B. Latitude of Town C
Town B is located at (20°N, 20°E). If the plane flies due north to Town C, we are only interested in the latitude of Town C.
Since the plane flies directly north from Town B, it will retain the same longitude (20°E) while moving to a different latitude.
Given that Town C is directly north of Town B, the latitude of Town C would increase. However, since no specific distance or endpoint is provided for the flight to Town C, it is not possible to calculate an exact latitude. We can only state that Town C is located at 20°N plus the distance flown north.
If we assume Town C is moving northward indefinitely, we would say that Town C's latitude increases but does not have a specific end point. Therefore, without additional information about the distance moved northward, we cannot provide a numerical answer.
In conclusion, the latitude of Town C remains undetermined without more specific information about how far north it travels:
\[
\text{Latitude of Town C = unknown}
\]
If you have further distance or specific information about Town C, please provide it for a more exact value.
### A. Calculating the average speed of the aeroplane
1. **Determine the distance between Town A and Town B.**
The coordinates of Town A are (20°N, 60°E) and Town B are (20°N, 20°E). Both towns are situated at the same latitude (20°N), so the distance between them can be calculated as the difference in longitude.
\[
\text{Longitude difference} = 60°E - 20°E = 40°
\]
To convert this angular distance to kilometers, we can use the formula for the distance along a circle of latitude:
\[
\text{Distance} = \text{(Longitude difference in degrees)} \times \left( \frac{ \pi \times R \times \cos(\text{latitude in radians})}{180} \right)
\]
where \( R \) is the average radius of the Earth (approximately 6371 km), and latitude in radians is \( 20° \times \frac{\pi}{180} \).
\[
\text{Distance} = 40° \times \left( \frac{ \pi \times 6371 \times \cos(20° \times \frac{\pi}{180})}{180} \right)
\]
Calculating \( \cos(20°) \):
\[
\cos(20°) \approx 0.9397
\]
Substituting values:
\[
\text{Distance} \approx 40 \times \frac{ \pi \times 6371 \times 0.9397}{180}
\]
\[
\text{Distance} \approx 40 \times \frac{20021.3 \times 0.9397}{180} \approx 40 \times 109.835 \approx 4393.4 \, \text{km}
\]
2. **Calculate average speed.**
The journey takes 6 hours, so the average speed \( v \) can be calculated as:
\[
v = \frac{\text{Distance}}{\text{Time}} = \frac{4393.4 \, \text{km}}{6 \, \text{hours}} \approx 732.2 \, \text{km/h}
\]
Correct to three significant figures, the average speed is:
\[
\boxed{732 \, \text{km/h}}
\]
### B. Latitude of Town C
Town B is located at (20°N, 20°E). If the plane flies due north to Town C, we are only interested in the latitude of Town C.
Since the plane flies directly north from Town B, it will retain the same longitude (20°E) while moving to a different latitude.
Given that Town C is directly north of Town B, the latitude of Town C would increase. However, since no specific distance or endpoint is provided for the flight to Town C, it is not possible to calculate an exact latitude. We can only state that Town C is located at 20°N plus the distance flown north.
If we assume Town C is moving northward indefinitely, we would say that Town C's latitude increases but does not have a specific end point. Therefore, without additional information about the distance moved northward, we cannot provide a numerical answer.
In conclusion, the latitude of Town C remains undetermined without more specific information about how far north it travels:
\[
\text{Latitude of Town C = unknown}
\]
If you have further distance or specific information about Town C, please provide it for a more exact value.
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