Difference between revisions of "2018 AMC 12B Problems/Problem 23"
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== Problem == | == Problem == | ||
− | Ajay is standing at point <math>A</math> near Pontianak, Indonesia, <math>0^\circ</math> latitude and <math>110^\circ \text{ E}</math> longitude. Billy is | + | Ajay is standing at point <math>A</math> near Pontianak, Indonesia, <math>0^\circ</math> latitude and <math>110^\circ \text{ E}</math> longitude. Billy is standing at point <math>B</math> near Big Baldy Mountain, Idaho, USA, <math>45^\circ \text{ N}</math> latitude and <math>115^\circ \text{ W}</math> longitude. Assume that Earth is a perfect sphere with center <math>C</math>. What is the degree measure of <math>\angle ACB</math>? |
<cmath>\textbf{(A) }105 \qquad | <cmath>\textbf{(A) }105 \qquad | ||
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Suppose that Earth is a unit sphere with center <math>(0,0,0).</math> We can let | Suppose that Earth is a unit sphere with center <math>(0,0,0).</math> We can let | ||
− | <cmath>A=(1,0,0), B=\left(-\frac{1}{2},\frac{1}{2},\frac{\sqrt 2}{2}\right).</cmath>The angle <math>\theta</math> between these two vectors satisfies <math>\cos\theta=A\cdot B=-\frac{1}{2},</math> yielding <math>\theta=120^{\circ},</math> or <math>\boxed{\textbf{C}.}</math> | + | <cmath>A=(1,0,0), B=\left(-\frac{1}{2},\frac{1}{2},\frac{\sqrt 2}{2}\right).</cmath>The angle <math>\theta</math> between these two vectors satisfies <math>\cos\theta=A\cdot B=-\frac{1}{2},</math> yielding <math>\theta=120^{\circ},</math> or <math>\boxed{\textbf{C}}.</math> |
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+ | Note (not by author): Alternatively, to find the angle without dot products, one may compute the distance from <math>A</math> to <math>B</math> as | ||
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+ | <math>\sqrt{\left(\frac{3}{2}\right)^2+\left(\frac{1}{2}\right)^2+\left(\frac{\sqrt{2}}{2}\right)^2}=\sqrt{3}</math>. | ||
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+ | From the Law of Cosines, <math>3=1^2+1^2-2\cos{\theta}</math>, so <math>\cos{\theta}=-\frac{1}{2},</math> from which the desired conclusion follows. | ||
==See Also== | ==See Also== | ||
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{{AMC12 box|year=2018|ab=B|num-b=22|num-a=24}} | {{AMC12 box|year=2018|ab=B|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
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+ | [[Category:Intermediate Geometry Problems]] |
Latest revision as of 01:27, 24 February 2021
Problem
Ajay is standing at point near Pontianak, Indonesia, latitude and longitude. Billy is standing at point near Big Baldy Mountain, Idaho, USA, latitude and longitude. Assume that Earth is a perfect sphere with center . What is the degree measure of ?
Solution
Suppose that Earth is a unit sphere with center We can let The angle between these two vectors satisfies yielding or
Note (not by author): Alternatively, to find the angle without dot products, one may compute the distance from to as
.
From the Law of Cosines, , so from which the desired conclusion follows.
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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