Exercises: 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.

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

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?

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.

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?

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?

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

In a Geometry exam, Cian has to calculate the distance between two marine buoys located on the same terrestrial meridian ($0^\circ$, the Greenwich meridian): buoy Alpha, located at $42^\circ\text{ N}$, and buoy Beta, located at $18^\circ\text{ S}$. Cian writes on his answer sheet:

«Since both buoys are on the Greenwich meridian, the angular difference between them is $\alpha = 42^\circ - 18^\circ = 24^\circ$. Therefore, multiplying by the theoretical distance per degree of meridian, the distance is $d = 24 \times 111.2 = 2668.8\text{ km}$».

1. a) Locate and explain, with reasoning, Cian's conceptual error, based on the geometric definition of the equator as the reference for latitude.
2. b) Correct the procedure and find the actual correct distance between the two buoys along the meridian.

Two Spanish cities, Santander ($43.5^\circ\text{ N}$) and Jaén ($37.8^\circ\text{ N}$), are located almost exactly on the same terrestrial meridian ($3.8^\circ\text{ W}$).

1. a) Estimate: Without using a calculator or decimals, round both latitudes to the nearest whole number. Mentally calculate the difference in degrees and estimate the distance in kilometres knowing that each degree of meridian is approximately equivalent to $110\text{ km}$.
2. b) Calculate: Now carry out the rigorous calculation using the given actual decimal values ($43.5^\circ$ and $37.8^\circ$) and the theoretical equivalence of $111.2\text{ km}$ per degree.
3. c) Check: Find the difference in kilometres between your estimate and the accurate calculation. Do you consider that the initial estimate was accurate enough for a transport company to evaluate fuel consumption? Justify your answer.

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.

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 kilometres that the whale travels on its outward journey.

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.

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 ($180^\circ$).

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).

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?

An oceanographic research buoy belonging to the Marine Institute is initially anchored in the Atlantic Ocean at $44^\circ\text{ N}$ and $5^\circ\text{ W}$. During a severe storm, it breaks free from its moorings and drifts in a straight line heading south along its meridian ($5^\circ\text{ W}$). Weeks later, the satellite locates the buoy after it has traveled exactly $5448.8\text{ km}$.

1. a) Knowing that each degree of a meridian corresponds to $111.2\text{ km}$, find the angular difference $\alpha$ traveled by the buoy.
2. b) Deduce the exact final geographic coordinates (latitude and longitude) where the buoy is located and justify in which hemisphere it has come to a stop.
3. c) Formulate a general equation where the unknown is the final latitude $x$ that allows solving this problem when starting from a northern latitude $\text{Lat}_0$ and moving southwards past the equator by a given distance $d$.

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 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 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 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 $12.50 \text{ €/m}^2$. 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?

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

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

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

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

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.

An air transport company designs a route between Tromsø (Norway, located at $70^\circ\text{ N}, 19^\circ\text{ E}$) and a research station in Point Hope (Alaska, located at $70^\circ\text{ N}, 161^\circ\text{ W}$). Note that the meridians $19^\circ\text{ E}$ and $161^\circ\text{ W}$ are antimeridians (they sum to $180^\circ$), so together they form a complete great circle passing through both poles. Two routes following great-circle arcs are proposed:

• Polar Method: Fly north along the meridian $19^\circ\text{ E}$ to the North Pole ($90^\circ\text{ N}$) and continue descending along the meridian $161^\circ\text{ W}$ to $70^\circ\text{ N}$.

• Equatorial Method: Fly south along the meridian $19^\circ\text{ E}$ to the equator ($0^\circ$), travel half the equator ($180^\circ$ of longitude along the equatorial circle, whose radius equals that of the meridian) to the meridian $161^\circ\text{ W}$, and climb north to $70^\circ\text{ N}$.

1. a) Calculate the distance in kilometers traveled following the Polar Method using the equivalence of $111.2\text{ km}$ per degree.
2. b) Calculate the distance in kilometers traveled following the Equatorial Method.
3. c) Compare both methods by quantifying the difference in kilometers and justify geometrically why intercontinental commercial flights between high latitudes of the northern hemisphere choose polar routes instead of descending toward more temperate latitudes.

Two oceanographic research vessels sail along the same Atlantic meridian ($15^\circ\text{ W}$). Vessel A departs from the point ($44^\circ\text{ N}, 15^\circ\text{ W}$) heading south, and Vessel B departs from the point ($10^\circ\text{ S}, 15^\circ\text{ W}$) heading north. Both begin their journey simultaneously at 08:00 (solar time of their meridian) at an identical constant speed of $27.8\text{ km/h}$.

1. a) Model the initial distance in kilometers separating both vessels (considering $111.2\text{ km}$ per degree of meridian) and calculate their combined approach speed expressed both in $\text{km/h}$ and in degrees of latitude per hour.
2. b) How many hours will it take for them to meet, and what will be the exact geographic coordinates (latitude and longitude) of the meeting point? Reason whether that point is in the northern or southern hemisphere.
3. c) When the meeting occurs, they send a radio report to a receiving base located at longitude $75^\circ\text{ E}$. Determine the theoretical solar time difference between the meeting place and the receiving base, and deduce the theoretical solar time at the receiving base at the moment they receive the message.

Topic challenge. Design an emergency bioclimatic shelter for a nature reserve in your county.

1. Location: Go to the Ordnance Survey Ireland (OSI) viewer or Google Earth, locate your chosen reserve and write down its exact geographical coordinates (latitude and longitude).

2. Geometric structure: Design the shelter by combining a polyhedron (habitable base module, for example a right prism) and a solid of revolution (hemispherical roof or cylindrical water tank). Indicate on your sketch at least one pair of parallel planes and two perpendicular edges key to its stability.

3. Calculation with a spreadsheet: Measure a household container to use as a $1:20$ scale model or define real dimensions in metres. In a spreadsheet, programme the formulas to find the surface area of the polyhedron (required thermal insulation) and the total volume of the resulting composite geometric solid.

4. Final proposal: Write a brief report justifying the feasibility of the shelter, its habitable capacity in $\text{m}^3$ and the waterproofing cost at a rate of $18.50\text{ €/m}^2$.

Evaluation criteria:

• Rigorous determination of latitude and longitude, and correct identification of parallelism and perpendicularity.

• Precision in calculating the surface area of the polyhedron and the volume of the composite solid.

• Correct automation of formulas in the spreadsheet using the decimal point.

• Quality and coherence of the submitted technical report.