Latitude and Longitude

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

1. Equator.

2. Greenwich Meridian.

3. North Pole.

4. Any parallel.

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

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

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

Answer with reasoning:

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

State in which hemisphere (North or South for latitude, East or West for longitude) the following geographical points are located:

• 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 a 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.

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

If you travel by airplane from Chicago ($40^\circ$ 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 the latitude of that point be? (Same value but opposite direction).
2. b) If the longitude of a point and its antipode must add up to $180^\circ$ and be in opposite hemispheres, what would the longitude of that point be?

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

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

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

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

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

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

Consider the Earth to be a perfect sphere of radius $6371 \text{ mi}$. Express the result in miles. Does the traveler move along a meridian or a parallel? Justify your answer.

For a school project, you need to calculate the length of the $40^\circ$ N parallel of latitude. Knowing that the Earth's radius is $R \approx 6\,371$ miles, calculate:

1. a) The radius of the circle formed by this parallel (use the expression $r = R \cdot \cos(\text{latitud})$).
2. b) The total length (perimeter) of this parallel.

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

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

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 cities, A and B, are located on the same parallel at a latitude of $45^\circ \text{ N}$. City A has a longitude of $20^\circ \text{ E}$ and city B a longitude of $70^\circ \text{ E}$.

Knowing that the radius of the Earth is approximately $6371 \text{ mi}$:

1. Calculate the radius of the $45^\circ \text{ N}$ parallel (Remember that the radius of a parallel at a latitude $\alpha$ is calculated as $r = R \cdot \cos \alpha$. For $45^\circ$, you can use $0,707$ as an approximation of the cosine).

2. Determine the distance in miles that separates both cities following the arc of the parallel.

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

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

1. a) Calculate how many miles a nautical mile is equivalent to (Radius of the Earth = $6\,371$ miles).
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?

A submarine travels along the seabed from a point X $(0^\circ, 175^\circ \text{ E})$ to a point Y $(0^\circ, 170^\circ \text{ O})$.

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

2. If the submarine travels at a constant speed of $20 \text{ nudos}$ ($1 \text{ nudo} \approx 1,85 \text{ mph}$), how long will it take to complete the journey? (Radius of the Earth: $6371 \text{ mi}$).

3. Was there a change of date during the journey? Explain why.

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

1. a) How many degrees of longitude correspond to a one-hour difference?
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 a live space rocket launch from Cape Canaveral (longitude $80^\circ$ $36'$ W). If the launch is scheduled for 2:30 p.m. local time in Florida, and knowing that each $15^\circ$ time zone is equivalent to a theoretical one-hour difference, at what time should they tune into the live stream in Spain? Use the theoretical time zone diagram to carry out the calculation.

An airplane takes off from Tokyo ($35^\circ$ N, $139^\circ$ E) at 8:00 a.m. 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 airplane lands? Take into account the date change when crossing the Greenwich antimeridian ($180^\circ$).

A reconnaissance plane takes off from a base located at coordinates $(30^\circ \text{ N}, 15^\circ \text{ O})$ at 8:00 a.m. local time. It flies directly East following the same parallel until it reaches a city located at $60^\circ$ longitude East. The flight has a total duration of 9 hours.

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

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

1. Determine the number of faces, vertices and edges of the resulting polyhedron and check if 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 water tank has the shape of a right regular hexagonal prism. It is known that the radius of the circumscribed circle of 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 (bottom base) if the price of the material is $12,50 \text{ \$/m}^2$. Round the result to two decimal places.

A perfume company designs a bottle in the shape of an inverted frustum of a cone topped by a hemisphere that fits perfectly on the top base. The design data is as follows:

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

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

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

Calculate the total volume of the bottle and determine if it is able to hold a standard $330 \text{ ml}$ bottle of essence. (Take $\pi \approx 3,14$).

There is a cylinder of radius $R$ and height $2R$. A sphere of radius $R$ is placed inside it, fitting tangentially with 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 is left free inside the cylinder once the sphere is inserted?