An aircraft flies from point X (32° N, 28° E) to point Y (32° N, 22° W) with a speed of 1250 km/h. It then flies from Y to another point Z (18° N, 22° W) with a speed of 823 km/h.... An aircraft flies from point X (32° N, 28° E) to point Y (32° N, 22° W) with a speed of 1250 km/h. It then flies from Y to another point Z (18° N, 22° W) with a speed of 823 km/h. Calculate: a. The distance XY b. The distance YZ c. The total time taken to travel from X to Z
Understand the Problem
The question describes an aircraft flying between three points (X, Y, and Z) with given coordinates and speeds for each leg of the journey. The problem asks to calculate the distance between points X and Y, the distance between points Y and Z, and the total time taken for the entire journey from X to Z.
Answer
a. $5561.75 \, km$ b. $1543.31 \, km$ c. $6.33 \, hours$
Answer for screen readers
a. The distance XY is approximately $5561.75 , km$. b. The distance YZ is approximately $1543.31 , km$. c. The total time taken to travel from X to Z is approximately $6.33 , hours$.
Steps to Solve
- Calculate the distance XY
Since X and Y are at the same latitude (32°N), the distance between them can be calculated using the longitude difference. The longitude of X is 28°E and the longitude of Y is 22°W. The difference in longitude is $28° + 22° = 50°$. The formula for the distance between two points on the same latitude is:
$Distance = \frac{2 \pi r \Delta Longitude}{360}$
Where $r$ is the radius of the Earth (approximately 6371 km) and $\Delta Longitude$ is the difference in longitude in degrees. Thus the distance between X and Y is:
$Distance_{XY} = \frac{2 \pi (6371 , km) (50°)}{360°} \approx 5561.75 , km $
- Calculate the distance YZ
Since Y and Z are on the same longitude (22°W), the distance between them can be calculated using the latitude difference. The latitude of Y is 32°N and the latitude of Z is 18°N. The difference in latitude is $32° - 18° = 14°$. The formula for the distance between two points on the same longitude is:
$Distance = \frac{2 \pi r \Delta Latitude}{360}$
Where $r$ is the radius of the Earth (approximately 6371 km) and $\Delta Latitude$ is the difference in latitude in degrees. Thus the distance between Y and Z is:
$Distance_{YZ} = \frac{2 \pi (6371 , km) (14°)}{360°} \approx 1543.31 , km$
- Calculate the time taken to travel from X to Y
We know the distance $Distance_{XY} \approx 5561.75 , km$ and the speed $Speed_{XY} = 1250 , km/h$. $Time = \frac{Distance}{Speed}$, so
$Time_{XY} = \frac{5561.75 , km}{1250 , km/h} \approx 4.45 , hours$
- Calculate the time taken to travel from Y to Z
We know the distance $Distance_{YZ} \approx 1543.31 , km$ and the speed $Speed_{YZ} = 823 , km/h$. $Time = \frac{Distance}{Speed}$, so
$Time_{YZ} = \frac{1543.31 , km}{823 , km/h} \approx 1.88 , hours$
- Calculate the total time taken to travel from X to Z
The total time is the sum of the time taken to travel from X to Y and the time taken to travel from Y to Z.
$TotalTime = Time_{XY} + Time_{YZ} \approx 4.45 , hours + 1.88 , hours \approx 6.33 , hours$
a. The distance XY is approximately $5561.75 , km$. b. The distance YZ is approximately $1543.31 , km$. c. The total time taken to travel from X to Z is approximately $6.33 , hours$.
More Information
The calculations assume a perfectly spherical Earth. In reality, the Earth is an oblate spheroid, and more complex calculations would be needed for higher accuracy.
Tips
A common mistake is to forget to convert the angular difference (in degrees) to a distance using the Earth's radius. Also, students may incorrectly add longitudes if they are in the same direction (both East or both West), or incorrectly subtract if they are in different directions. Another common mistake is using the incorrect speed when calculating the time for each leg, or forgetting to sum the times for each leg to get the total time Not using the correct formula is also a common mistake
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