2023 AMC 10B Problems/Problem 20
Contents
Problem
Four congruent semicircles are drawn on the surface of a sphere with radius , as
shown, creating a close curve that divides the surface into two congruent regions.
The length of the curve is
. What is
?
Solution 1
There are four marked points on the diagram; let us examine the top two points and call them and
. Similarly, let the bottom two dots be
and
, as shown:
This is a cross-section of the sphere seen from the side. We know that , and by Pythagorean therorem,
Each of the four congruent semicircles has the length as a diameter (since
is congruent to
and
), so its radius is
Each one's arc length is thus
We have of these, so the total length is
, so thus our answer is
~Technodoggo
Solution 2
Assume ,
,
, and
are the four points connecting the semicircles. By law of symmetry, we can pretty confidently assume that
is a square. Then,
, and the rest is the same as the second half of solution
.
~jonathanzhou18
Solution 3
We put the sphere to a coordinate space by putting the center at the origin.
The four connecting points of the curve have the following coordinates: ,
,
,
.
Now, we compute the radius of each semicircle.
Denote by the midpoint of
and
. Thus,
is the center of the semicircle that ends at
and
.
We have
.
Thus,
.
In the right triangle , we have
.
Therefore, the length of the curve is
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 4
Note that each of the diameters are the chord of the sphere of a quarter arc. Thus, the semicircles diameter's length is . Thus, the entire curve is
. Therefore, the answer is
.
~andliu766
Video Solution 1 by OmegaLearn
See also
2023 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 10 Problems and Solutions |
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