Task: Determine the distance that a ship sailing along a meridian from its home port to a destination must travel, if the home port is located on the equator and the destination is situated at a latitude of $10^\circ$.
Alice solved the task in the following steps:
(1) Alice claimed that for the purposes of this task, latitude can be considered the measure of the angle between the plane of the equator and the line $p$ passing through the Earth's center and the point of the ship’s destination on Earth's surface. She sketched a diagram of a ($2D$) cross-section of Earth through the plane containing line $p$, which is perpendicular to the plane of the equator. She aimed to find the length of the arc $\widehat{AB}$ corresponding to the central angle $\varphi=10^\circ$ (see the diagram).
(2) She expressed the measure of the angle $\varphi$ in terms of arc length: The given latitude of $10^\circ$ corresponds to $\frac{1}{18}$ of a straight angle, meaning: $$\varphi=\frac{1}{18}\pi\ \mathrm{(rad)}$$
(3) Alice claimed that in the circle representing Earth, the arc length corresponding to the central angle $\varphi=\frac{1}{18}\pi$ is: $$\frac{1}{18}\pi\approx0{.}175$$ (4) Alice added the unit and concluded that the sought distance the ship must travel is approximately $0{.}175\,\mathrm{km}$.
The result is incorrect. In which step did Alice make a mistake?
The mistake is in step (1). Alice misunderstood latitude. Latitude is the angular distance of a place on Earth's surface from the poles (in our case, from the North Pole). The angle $\varphi$ in the diagram should be $80^\circ$.
The mistake is in step (2). The given latitude of $10^\circ$ corresponds to $\frac{1}{10}$ of a straight angle.
The mistake is in step (3). In a circle with radius $r$, the arc length corresponding to the central angle $\varphi$ is $\varphi\cdot r$.
The mistake is in step (4). The calculated value of $0{.}175$ corresponds to $175\,\mathrm{km}$.
Alice made a mistake in step (3). The arc length $\frac{1}{18}\pi\approx 0{.}175$ corresponds to the central angle $\varphi=\frac{1}{18}\pi$ only in the unit circle. Hence, the result would be correct if Earth’s radius was $1\,\mathrm{km}$. In a circle with radius $r$, the arc length corresponding to the central angle $\varphi$ is $\varphi\cdot r$. The radius of Earth is $6\, 371\,\mathrm{km}$. Therefore, the arc length on Earth’s surface corresponding to the central angle $\varphi=\frac{1}{18}\pi$ is: $$\frac{\pi}{18}\cdot6\,371=1\,112\,\mathrm{km}.$$ Thus, a ship sailing along a meridian from a home port at a latitude of $10^\circ$ to a destination on the equator must travel approximately $1\,112\,\mathrm{km}$.
