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Difference between revisions of "2023 AMC 10B Problems/Problem 20"

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~jonathanzhou18
 
~jonathanzhou18
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==Solution 3==
 +
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We put the sphere to a coordinate space by putting the center at the origin.
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The four connecting points of the curve have the following coordinates: <math>A = \left( 0, 0, 2 \right)</math>, <math>B = \left( 2, 0, 0 \right)</math>, <math>C = \left( 0, 0, -2 \right)</math>, <math>D = \left( -2, 0, 0 \right)</math>.
 +
 +
Now, we compute the radius of each semicircle.
 +
Denote by <math>M</math> the midpoint of <math>A</math> and <math>B</math>. Thus, <math>M</math> is the center of the semicircle that ends at <math>A</math> and <math>B</math>.
 +
We have <math>M = \left( 1, 0, 1 \right)</math>.
 +
Thus, <math>OM = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}</math>.
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In the right triangle <math>\triangle OAM</math>, we have <math>MA = \sqrt{OA^2 - OM^2} = \sqrt{2}</math>.
 +
 +
Therefore, the length of the curve is
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<cmath>
 +
\begin{align*}
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4 \cdot \frac{1}{2} 2 \pi \cdot MA
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= \pi \sqrt{32} .
 +
\end{align*}
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</cmath>
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Therefore, the answer is \boxed{\textbf{(A) 32}}.
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 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Revision as of 17:14, 15 November 2023

Problem 20

Four congruent semicircles are drawn on the surface of a sphere with radius 2, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is 𝑛?

Solution 1

There are four marked points on the diagram; let us examine the top two points and call them $A$ and $B$. Similarly, let the bottom two dots be $C$ and $D$, as shown:

[asy] import graph; import geometry; unitsize(1cm); pair A = (-1.41, 1.41); pair B = (1.41, 1.41); pair C = (1.41, -1.41); pair D = (-1.41, -1.41); pair O = (0, 0); draw(circle(O,2)); draw(A--O--B,black+dashed); draw(C--O--D,black+dashed); dot(A);dot(B);dot(C);dot(D);dot(O); label("$A$", A, NW); label("$B$", B, NE); label("$C$", C, SE); label("$D$", D, SW); label("$O$", (0,0.1), N); [/asy]

This is a cross-section of the sphere seen from the side. We know that $\overline{AO}=\overline{BO}=\overline{CO}=\overline{DO}=2$, and by Pythagorean therorem, $\overline{AB}=2\sqrt2.$

Each of the four congruent semicircles has the length $AB$ as a diameter (since $AB$ is congruent to $BC,CD,$ and $DA$), so its radius is $\dfrac{2\sqrt2}2=\sqrt2.$ Each one's arc length is thus $\pi\cdot\sqrt2=\sqrt2\pi.$

We have $4$ of these, so the total length is $4\sqrt2\pi=\sqrt{32}\pi$, so thus our answer is $\boxed{\textbf{(A) }32.}$

~Technodoggo

Solution 2

Assume $A$, $B$, $C$, and $D$ are the four points connecting the semicircles. By law of symmetry, we can pretty confidently assume that $ABCD$ is a square. Then, $\overline{AB} = 2\sqrt2.$, and the rest is the same as the second half of solution $1$.

~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: $A = \left( 0, 0, 2 \right)$, $B = \left( 2, 0, 0 \right)$, $C = \left( 0, 0, -2 \right)$, $D = \left( -2, 0, 0 \right)$.

Now, we compute the radius of each semicircle. Denote by $M$ the midpoint of $A$ and $B$. Thus, $M$ is the center of the semicircle that ends at $A$ and $B$. We have $M = \left( 1, 0, 1 \right)$. Thus, $OM = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}$.

In the right triangle $\triangle OAM$, we have $MA = \sqrt{OA^2 - OM^2} = \sqrt{2}$.

Therefore, the length of the curve is \begin{align*} 4 \cdot \frac{1}{2} 2 \pi \cdot MA = \pi \sqrt{32} . \end{align*}

Therefore, the answer is \boxed{\textbf{(A) 32}}.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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