Question

Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ$ E longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ$ N latitude and $115^\circ$ W longitude. Assume that Earth is a perfect sphere with center $C.$ What is the degree measure of $\angle ACB?$

$105$

$112\tfrac{1}{2}$

$120$

$135$

$150$

Answer: C

Concepts:3D geometry sphere vector

Difficulty rating: 2400

Solution:

The longitudes differ by $360^\circ-(110^\circ+115^\circ)=135^\circ,$ and $B$ is at latitude $45^\circ$ N. Place $A=(1,0,0)$ on the unit sphere.

Then $B=\tiny\left(\cos45^\circ\cos135^\circ,\ \cos45^\circ\sin135^\circ,\ \sin45^\circ\right)$ $=\left(-\tfrac12,\tfrac12,\tfrac{\sqrt2}{2}\right).$ The dot product is $A\cdot B=-\tfrac12,$ so $\cos\angle ACB=-\tfrac12$ and $\angle ACB=120^\circ.$

Thus, the correct answer is C.

2018 AMC 12B Problem 23

Answer: C

Difficulty rating: 2400

Solution:

Problem 23 in Other Years