2018 AMC 12B Problem 23

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

The longitude difference is

360^\circ-(110^\circ+115^\circ)=135^\circ

Put both points on a unit sphere and take the dot product of their position vectors

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.

Problem 23 in other years