Solving Algebra Problem: Walking on the Earth's Surface

• analyzer
In summary, Arnoldo Téllez walked one mile to the south, then one mile to the east, and then one mile to the north, getting back to the point where he started. He could have started in the north pole, but he didn't.
analyzer

Homework Statement

Arnoldo Téllez walked one mile to the south, then one mile to the east, and then one mile to the north, getting back to the point where he started. He could have started in the north pole, but he didn't. Where did he start?
(Taken from ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY: A PROBLEM-SOLVING APPROACH, by Varberg and Fleming.)

The Attempt at a Solution

He could have started at several different points:
- At any of the points that are one mile to the north from the parallel whose length is one mile (if that parallel exists).
- At any of the points that are one mile to the north from the parallel whose length is one half of a mile (if that parallel exists), thus walking twice over that parallel.
- At any of the points that are one mile to the north from the parallel whose length is one third of a mile (if that parallel exists), thus walking three times over that parallel.
- and so on...

Yes. The parallels specified do exist, with points available one mile to the north - where?

Where?

Those points must be part of other parallels, respectively.

Sorry, I wasn't clear. Where are the parallels of the required length, given that they must also have points one mile to the north?

Those parallels must be on the southern hemispere. I don't know exactly where.

Where do you find such short parallels?

South Pole?

Near the South Pole, that is.

Correct! Very near the south pole, in fact. You can essentially ignore the curvature of the Earth to get a good approximation of how far they are from the pole...

The Antarctic?

Arnoldo Téllez could have started at any of the points which are one mile to the north from the parallel which is at 1/(2*pi) miles from the South Pole; i.e. he could have started at any of the points which are (1 + 1/(2*pi)) miles from the South Pole.

Right?

Sorry analyzer - yes, absolutely correct, for the "once round the pole" version...

1. How is algebra used in solving problems related to walking on the Earth's surface?

Algebra is used to calculate distances, speeds, and time taken to travel on the Earth's surface. It can also be used to determine the angle of elevation or depression when walking on different terrains.

2. What are some common algebraic equations used in solving problems related to walking on the Earth's surface?

Some common equations include the distance formula (d = rt), the Pythagorean theorem (a² + b² = c²), and the slope formula (m = Δy/Δx).

3. How can algebra help in determining the shortest path when walking on the Earth's surface?

Algebra can be used to find the shortest path by calculating the distance between two points and comparing it to other possible paths. This can be done using the distance formula or by finding the minimum value of a quadratic equation.

4. Can algebra be used to solve problems related to walking on curved surfaces?

Yes, algebra can be used to solve problems related to walking on curved surfaces. It involves using more advanced equations such as the arc length formula (s = rθ) and the curvature formula (κ = 1/r).

5. How does algebra play a role in navigation and map reading when walking on the Earth's surface?

Algebra plays a crucial role in navigation and map reading by helping to calculate distances, angles, and coordinates. It also helps in understanding scale and proportional relationships on maps and in using compass directions and bearings to navigate.

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