IOAA 2015, Теория, кратки задачи, български — Задача 15

1st version

1 Determine coordinates of Lima and Yogyakarta. [15]

Assume that the equator is the $z = 0$ plane and the Earth is a unit sphere.

The Cartesian coordinates of the Lima is

The Cartesian coordinates of the Yogyakarta is

2 Determine the normal of the plane defined by Lima, Yogyakarta, and center of the Earth. [20]

The normal of the plane containing the two places and the center of the Earth is

3 Find the equation of the (shortest flight) path from Lima to Yogyakarta. [25]

Let the equation of the place be $ax + by + cz = 0$ or equivalently

where $a' = -3.16702272082062$ and $c' = 1.23335628599724$. For the sake of simplicity, from now let us denote $a'$ and $c'$ as $a$ and $c$, respectively.

Intersection of the Earth and the plane is the large circle $x^2 + (-ax - cz)^2 + z^2 = 1$ or equivalently

It is also the equation for the shortest flight path from Lima to Yogyakarta.

4 Determine the z-value of the southernmost point of the path (at the point, the discriminant of the equation

equals zero). [30]

The plane $z = m$, it intersects the large circle at at either at one or two points. The number of intersection point at the highest or at the lowest point is one, i.e. when the discriminant is zero.

The discriminant is

Therefore $D = 0$, if $m = \pm\sqrt{\frac{1 + a^2}{1 + a^2 + c^2}} = 0.937444937497383$. Then choose

5 Determine the latitude of the southernmost point. [10]

Thus, the latitude of the point is $\sin^{-1}z = 1.21521692653748 = 69.6268010651290^{\circ} = 69^{\circ}37'36''S$.

2nd version (using the Law of Cosines for sides)

Answer and Marking Scheme:

1 Determine length of the sides of the triangle [20]

$s_2 = 90^{\circ} - 12^{\circ}2' = 77^{\circ}58' = 1.3608\ rad, s_3 = 90^{\circ} - 7^{\circ}47' = 82^{\circ}13' = 1.4350\ rad; A_1 = (180^{\circ} - 77^{\circ}1') + (180^{\circ} - 110^{\circ}26') = 172^{\circ}33' = 3.0116\ rad$

Using Law of Cosines for sides

to obtain $s_1 = 2.7724.\mathrm{rad}$

2 Determine the angles of the triangle [30]

Then use

to obtain $A_2 = 0.35921$ rad. Similarly, use Law of Cosines

to get that $A_3 = 0.3642$ rad.

3 Use Law of Cosines for sides to construct an equation in $\cos s_2'$. [20]

If P is the southernmost point, then the angle $A_3$ is a right angle, $A_3 = \frac{\pi}{2}$, the Law of

Thus,

Then substitute it to get

4 Solve the equation [20]

Let $x = \cos s_2'$. Then

Multiply both sides by $x$ to get

which is a quadratic equation of $x^2$. The roots are $\pm0.13538$ and $\pm0.93739$.

5 Determine the Latitude [10]

Thus,

or

The latitude is $69^{\circ}37'S$ or 7°47′S. The second one is impossible because it higher then the latitude of Lima. Thus the answer is

3rd version (using Napier’s rule)

Answer and Marking Scheme:

1 Determine length of the sides of the triangle [20]

$s_2 = 90^{\circ} - 12^{\circ}2' = 77^{\circ}58' = 1.3608\ rad, s_3 = 90^{\circ} - 7^{\circ}47' = 82^{\circ}13' = 1.4350\ rad; A_1 = (180^{\circ} - 77^{\circ}1') + (180^{\circ} - 110^{\circ}26') = 172^{\circ}33' = 3.0116\ rad$

Using Law of Cosines for sides

to obtain $s_1 = 2.7724.\mathrm{rad}$

2 Determine the angles of the triangle [30]

Then use

to obtain $A_2 = 0.35921$ rad. Similarly, use Law of Cosines

to get that $A_3 = 0.3642$ rad.

3 Find the latitude of the point [50]

Let $P$ be the southernmost point on the flight path. Since the triangle has a right triangle at P, then use one of Napier’s rules

to get $s_2' = 0.35576\ rad = 20^{\circ}23'$. Therefore, the latitude of $P$ is