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.use the formula theta÷360 ×2π6400
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.use the formula theta÷360 ×2π6400
Answers
GPT-4o mini
To solve these problems, we will break them down step-by-step.
### A. Average Speed Calculation
**Step 1: Calculate the distance between Town A and Town B.**
Both towns are located at the same latitude (20°N). Thus, the distance between them can be calculated considering only their longitudinal difference.
- **Longitude of Town A:** 60°E
- **Longitude of Town B:** 20°E
**Longitude difference** = \( 60°E - 20°E = 40° \)
**Step 2: Calculate the distance corresponding to the longitudinal difference.**
The circumferential distance of the Earth at the equator is approximately \( 2\pi \times 6400 \text{ km} \). To find the distance for 1° of longitude:
\[
\text{Distance per degree} = \frac{2\pi \times 6400 \text{ km}}{360} \approx 111.32 \text{ km/degree}
\]
**Step 3: Calculate the distance for 40°.**
\[
\text{Distance} = 40° \times 111.32 \text{ km/degree} \approx 4452.8 \text{ km}
\]
**Step 4: Calculate the average speed.**
The journey takes 6 hours, so we can calculate the average speed (in km/h):
\[
\text{Average speed} = \frac{\text{Distance}}{\text{Time}} = \frac{4452.8 \text{ km}}{6 \text{ hours}} \approx 742.1333 \text{ km/h}
\]
Rounding this to three significant figures:
\[
\text{Average speed} \approx 742 \text{ km/h}
\]
### B. Latitude Calculation for Town C
**Step 1: Determine the direction of the flight from Town B to Town C.**
Town B is at \(20°N\). The plane flies due north to reach town C.
**Step 2: Calculate the latitude of Town C.**
Assume the plane flies a certain distance northward. We need to figure out the distance in terms of degrees.
If we denote \(d\) as the distance flown north in km, we can use the same formula to convert this distance into degrees of latitude:
\[
\text{Degrees} = \frac{d}{\text{Distance per degree of latitude}}
\]
The distance per degree of latitude is approximately 111.32 km/degree (this is relatively constant because lines of latitude are parallel).
**Step 3: Let’s calculate how far north the plane goes.**
We’re not given a specific distance so we can only express latitude as follows. If the plane travels \(d\) km north (whatever distance this may be):
\[
\text{Latitude change} = \frac{d}{111.32}
\]
Starting from \(20°N\):
\[
\text{Latitude of Town C} = 20°N + \frac{d}{111.32}
\]
Since no specific distance is provided for the northward journey from Town B to Town C, we cannot calculate the exact latitude of Town C numerically without that distance.
If you happen to have the distance \(d\) for the northward flight, please provide that, and I can calculate the latitude of Town C for you. Otherwise, the final latitude of Town C is expressed in terms of the distance flown north.
### A. Average Speed Calculation
**Step 1: Calculate the distance between Town A and Town B.**
Both towns are located at the same latitude (20°N). Thus, the distance between them can be calculated considering only their longitudinal difference.
- **Longitude of Town A:** 60°E
- **Longitude of Town B:** 20°E
**Longitude difference** = \( 60°E - 20°E = 40° \)
**Step 2: Calculate the distance corresponding to the longitudinal difference.**
The circumferential distance of the Earth at the equator is approximately \( 2\pi \times 6400 \text{ km} \). To find the distance for 1° of longitude:
\[
\text{Distance per degree} = \frac{2\pi \times 6400 \text{ km}}{360} \approx 111.32 \text{ km/degree}
\]
**Step 3: Calculate the distance for 40°.**
\[
\text{Distance} = 40° \times 111.32 \text{ km/degree} \approx 4452.8 \text{ km}
\]
**Step 4: Calculate the average speed.**
The journey takes 6 hours, so we can calculate the average speed (in km/h):
\[
\text{Average speed} = \frac{\text{Distance}}{\text{Time}} = \frac{4452.8 \text{ km}}{6 \text{ hours}} \approx 742.1333 \text{ km/h}
\]
Rounding this to three significant figures:
\[
\text{Average speed} \approx 742 \text{ km/h}
\]
### B. Latitude Calculation for Town C
**Step 1: Determine the direction of the flight from Town B to Town C.**
Town B is at \(20°N\). The plane flies due north to reach town C.
**Step 2: Calculate the latitude of Town C.**
Assume the plane flies a certain distance northward. We need to figure out the distance in terms of degrees.
If we denote \(d\) as the distance flown north in km, we can use the same formula to convert this distance into degrees of latitude:
\[
\text{Degrees} = \frac{d}{\text{Distance per degree of latitude}}
\]
The distance per degree of latitude is approximately 111.32 km/degree (this is relatively constant because lines of latitude are parallel).
**Step 3: Let’s calculate how far north the plane goes.**
We’re not given a specific distance so we can only express latitude as follows. If the plane travels \(d\) km north (whatever distance this may be):
\[
\text{Latitude change} = \frac{d}{111.32}
\]
Starting from \(20°N\):
\[
\text{Latitude of Town C} = 20°N + \frac{d}{111.32}
\]
Since no specific distance is provided for the northward journey from Town B to Town C, we cannot calculate the exact latitude of Town C numerically without that distance.
If you happen to have the distance \(d\) for the northward flight, please provide that, and I can calculate the latitude of Town C for you. Otherwise, the final latitude of Town C is expressed in terms of the distance flown north.