The U.S.S. Lollipop is on assignment in the Atlantic Ocean. It travels from a longitude of 70 degrees west to 20 degrees west, along the latitude of 40 degrees north. How far does it travel? (Assume that the radius of the Earth is 6,400 km.)
Erm I don't know how to solve this. I graphed it and it is basically a small line segment on a sphere, but how to solve it? should I draw the radius to connect it? thx
Along the equator, if you travel an angle of θ, your distance is rθ.
The travel is along the 40° latitude, so moving through an angle of θ° of longitude, you only travel rθcos40°.
The Lollipop moved 50°, so it traveled
(6400)(50*π/180)(cos40°) = 4278 km
thank you that helped !
Hi. I don't understand why you multiply by 50*pi/180. The 50 degrees is the difference in the longitude, but why isn't that divided by 360? Why the pi/180? If someone could clear that up I would appreciate it.
Sorry, never mind. I figured it out.
To find the distance traveled by the U.S.S. Lollipop, you can use the spherical law of cosines or the haversine formula. Let's use the haversine formula as it is more commonly used for this type of calculation.
The haversine formula allows us to calculate the great-circle distance between two points on a sphere using the latitude and longitude coordinates.
Here are the steps to follow:
1. Convert the given latitude and longitude coordinates from degrees to radians. This is because the trigonometric functions used in the haversine formula work with radians.
So, for the initial point at 70 degrees west and 40 degrees north, we convert it to (-70, 40) in radians as follows:
- Convert longitude: -70 degrees * (π/180) radians/degree = -1.222 radians
- Convert latitude: 40 degrees * (π/180) radians/degree = 0.698 radians
Similarly, for the final point at 20 degrees west and 40 degrees north, we convert it to (-20, 40) in radians:
- Convert longitude: -20 degrees * (π/180) radians/degree = -0.349 radians
- Convert latitude: 40 degrees * (π/180) radians/degree = 0.698 radians
2. Calculate the differences in latitude and longitude between the two points:
- Latitude difference: 0.698 - 0.698 = 0 radians
- Longitude difference: -0.349 - (-1.222) = 0.873 radians
3. Apply the haversine formula using the following formula:
a. Calculate the square of half the chord length between the latitude points:
h = sin²(Δlat/2) + cos(lat1) * cos(lat2) * sin²(Δlon/2)
b. Calculate the angular distance in radians using the inverse haversine (opposite of sine) of h:
d = 2 * R * atan2(sqrt(h), sqrt(1-h))
Where:
- Δlat = latitude difference
- Δlon = longitude difference
- R = radius of the Earth (given as 6,400 km)
4. Substitute the calculated values into the formula and solve for the distance traveled:
d = 2 * (6,400 km) * atan2(sqrt(sin²(0/2) + cos(0.698) * cos(0.698) * sin²(0.873/2))), sqrt(1 - sin²(0/2) + cos(0.698) * cos(0.698) * sin²(0.873/2)))
Using a scientific calculator or software, you can evaluate this expression and find the distance traveled by the U.S.S. Lollipop.
Note: The haversine formula assumes a perfect sphere, so the result might be slightly different from the actual distance due to the Earth's oblate spheroid shape. However, for most practical purposes, it provides a good approximation.
I hope this helps you solve the problem! Let me know if you have any further questions.