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 Madrid ($40^\circ$ 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, Villa Sol and Villa Luna, are located on the same meridian. If Villa Sol has a latitude of $45^\circ$ N and Villa Luna has a latitude of $20^\circ$ S, how many degrees of meridian arc separate them?

A humpback whale migrates along exactly the same meridian from a feeding area located at $65^\circ$ N latitude to a breeding area at $15^\circ$ S. If the mean radius of the Earth is taken to be $6\,371$ km, calculate the total distance in kilometers that the whale travels on its outward journey.

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 kilometers 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 traveled?

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.

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?

Calculate the shortest distance between two points, P and Q, located on the Earth's surface if their coordinates are:

• P: $(40^\circ \text{ N}, 10^\circ \text{ O})$

• Q: $(40^\circ \text{ S}, 10^\circ \text{ O})$

Assume that the Earth is a perfect sphere of radius $6371 \text{ km}$. Express the result in kilometers. Does the traveler travel along a meridian or along a parallel? Justify your answer.

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 in to 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 ($180^\circ$).

A reconnaissance aircraft takes off from a base located at coordinates $(30^\circ \text{ N}, 15^\circ \text{ O})$ at 08:00 local time. It flies directly East along the same parallel until it reaches a city located at $60^\circ$ East longitude. The flight has a total duration of 9 hours.

At what local time in the destination city will the aircraft land? Take into account the change in time zones during the journey (assume theoretical time zones of $15^\circ$ each).

A submarine travels along the seabed from a point X $(0^\circ, 175^\circ \text{ E})$ to a point Y $(0^\circ, 170^\circ \text{ O})$. The figure shows the Equator seen from the North Pole, with the two points marked.

1. a) How many degrees of longitude has it travelled if it has followed the shortest route along the Equator?

1. b) If the submarine travels at a constant speed of $20 \text{ nudos}$ ($1 \text{ nudo} \approx 1.85 \text{ km/h}$), how long will it take to complete the journey? (Radius of the Earth: $6371 \text{ km}$).

1. c) Was there any date change during the journey? Explain why.

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 perfume company designs a bottle in the shape of an inverted frustum of a cone topped by a hemisphere that fits perfectly onto the upper base. The design data are as follows:

• Height of the frustum of a cone: $8 \text{ cm}$.

• Radius of the lower base (where it rests): $3 \text{ cm}$.

• Radius of the upper base (where the hemisphere fits): $6 \text{ cm}$.

Calculate the total volume of the bottle and determine whether it can hold a standard bottle of $330 \text{ ml}$ of essence. (Take $\pi \approx 3.14$).

A water tank is in the shape of a right regular hexagonal prism. It is known that the radius of the circle circumscribing the hexagonal base measures $4 \text{ m}$ and that the height of the prism is three times the apothem of the base.

Calculate the cost of waterproofing the side walls and the bottom (lower base) if the price of the material is $750 \text{ \text{RD\$}\ /m}^2$. Round the result to two decimal places.

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?