Longitude and latitude

Imagine you are a beginner cartographer. On your first map, you must locate the most important reference lines on Earth. Look at the diagram and match each letter with its corresponding name:

1. Equator.

2. Greenwich Meridian.

3. North Pole.

4. Any parallel.

Look at the following flat representation (planisphere) of an ocean region. Identify the coordinates (Latitude and Longitude) of point $P$ marked with a star.

State which hemisphere (Northern or Southern for latitude, Eastern or Western for longitude) the following geographical points are located in:

• Point A: $34^\circ$ N, $120^\circ$ E

• Point B: $12^\circ$ S, $45^\circ$ W

• Point C: $89^\circ$ N, $5^\circ$ W

• Point D: $40^\circ$ S, $170^\circ$ E

An explorer has mistakenly written down the following coordinates in his diary. Explain why these coordinates are geographically impossible:

1. a) Latitude: $105^\circ$ S, Longitude: $20^\circ$ E.
2. b) Latitude: $45^\circ$ N, Longitude: $210^\circ$ W.

Two ships, Alfa and Beta, are sailing across the Atlantic Ocean. Their coordinates are:

• Ship Alfa: $20^\circ$ N, $30^\circ$ W

• Ship Beta: $20^\circ$ N, $60^\circ$ W

Answer giving reasons:

1. a) Are they located on the same parallel or on the same meridian?
2. b) Which of the two ships is closer to the Greenwich Meridian?

If you travel by plane from Dublin ($40^\circ$ N, $3^\circ$ W) to a city that is exactly at its antipodes (the opposite point on the Earth's sphere):

1. a) What would be the latitude of that point? (Same value but opposite direction).
2. b) If the longitudes of a point and its antipode must sum to $180^\circ$ and be in opposite hemispheres, what would be the longitude of that point?

The antipodes of a location are said to be the point on the Earth's surface that is diametrically opposite to it. If the coordinates of your city are $40^\circ$ $25'$ N latitude and $3^\circ$ $41'$ W longitude:

1. a) What would be the latitude of your antipodes?
2. b) What would be the longitude of your antipodes?

(Hint: For the latitude, change the hemisphere; for the longitude, find the meridian that completes $180^\circ$ and change the hemisphere E/W).

Two towns, Ballysun and Ballymoon, are located on the same meridian. If Ballysun has a latitude of $45^\circ$ N and Ballymoon has a latitude of $20^\circ$ S, how many degrees of meridian arc separate them?

In navigation, the "nautical mile" is used, defined as the arc length of a meridian of $1'$ (one minute of arc).

1. a) Calculate how many kilometres one nautical mile is equivalent to (Radius of the Earth = $6\,371$ km).
2. b) If a sailboat has sailed along a meridian from a latitude of $30^\circ$ N to $35^\circ$ N, how many nautical miles has it travelled?

Two cities, P and Q, are situated on the same meridian. City P is at $42^\circ$ N and city Q is at $15^\circ$ N. Another city, R, is situated on the same parallel as Q, but at a distance of $20^\circ$ longitude to the East.

Calculate, using $R = 6\,371$ km:

1. a) The distance between P and Q along the meridian.
2. b) The distance between Q and R along the parallel.

Which of the two distances is greater?

Two oceanographic research ships are located on the Equator line. Ship A is at a longitude of $12^\circ$ E and ship B is located at a distance of $2\,500$ km to the West of ship A. Determine the exact longitude of ship B, knowing that the radius of the Earth is $6\,371$ km.

We know that the Earth takes 24 hours to complete a rotation of $360^\circ$ on its axis.

1. a) How many degrees of longitude correspond to a time difference of one hour?
2. b) If it is 12:00 noon in a city located at $0^\circ$ longitude (Greenwich), what time will it be in a city located at $30^\circ$ E?

A group of friends in Madrid (longitude $3^\circ$ $41'$ W) wants to watch live the launch of a space rocket from Cape Canaveral (longitude $80^\circ$ $36'$ W). If the launch is scheduled for 14:30 Florida local time, and knowing that each time zone of $15^\circ$ is equivalent to a theoretical time difference of one hour, at what time should they tune into the live broadcast in Spain? Use the theoretical time zone diagram to perform the calculation.

An aircraft takes off from Tokyo ($35^\circ$ N, $139^\circ$ E) at 08:00 on Monday bound for San Francisco ($37^\circ$ N, $122^\circ$ W). If the flight lasts 10 hours, what time and day will it be in San Francisco when the aircraft lands? Take into account the date change when crossing the Greenwich antimeridian ($180^\circ$).

Astronomical "solar noon" occurs when the Sun crosses the exact meridian of the location. The Earth completes a rotation of $360^\circ$ in 24 hours, meaning that each theoretical time zone of $15^\circ$ in longitude corresponds to 1 hour (or $1^\circ$ to 4 minutes).

1. a) Write the mathematical formula relating the angular difference in longitude $\Delta \lambda$ (in degrees) between two locations and the time difference between their respective solar noons $\Delta t$ (in minutes).
2. b) Dublin is located at $4.2^\circ\text{ E}$ longitude and Valentia Island (Kerry) is at $9.3^\circ\text{ W}$. Model and calculate the total angular difference between both locations, and determine the exact solar time difference in minutes.
3. c) If in Dublin solar noon occurs at 13:10 according to official winter clock time, at what official clock time does solar noon occur at Valentia Island? From the result, interpret why people in Kerry perceive that night falls "very late" relative to clock time.

A truncated octahedron is constructed from a regular octahedron with an edge length of $12 \text{ cm}$. To do this, each of the six vertices is cut off so that small square-based pyramids are removed, converting each original triangular face into a regular hexagon.

1. Determine the number of faces, vertices and edges of the resulting polyhedron and check whether it satisfies Euler's formula.

2. If the edge of the new polyhedron (that of the resulting hexagon) is $4 \text{ cm}$, calculate the total surface area of the truncated octahedron.

A cylinder has a radius of $R$ and a height of $2R$. Inside it, a sphere of radius $R$ is placed, which fits tangentially against the walls and bases of the cylinder.

1. Prove analytically the relationship between the volume of the sphere and the volume of the cylinder.

2. If the volume of the cylinder is $54 \text{ cm}^3$, what volume of air remains free inside the cylinder once the sphere is inserted?