Difference between revisions of "2005 AMC 12B Problems/Problem 16"

Latest revision as of 03:54, 16 January 2023

Problem

Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?

$\mathrm {(A)}\ \sqrt{2} \qquad \mathrm {(B)}\ \sqrt{3} \qquad \mathrm {(C)}\ 1+\sqrt{2}\qquad \mathrm {(D)}\ 1+\sqrt{3}\qquad \mathrm {(E)}\ 3$

Solution

The eight spheres are formed by shifting spheres of radius $1$ and center $(0, 0, 0)$ $\pm 1$ in the $x, y, z$ directions. Hence, the centers of the spheres are $(\pm 1, \pm 1, \pm 1)$ . For a sphere centered at the origin to contain all eight spheres, its radius must be greater than or equal to the longest distance from the origin to one of these spheres. This length is the sum of the distance from $(\pm 1, \pm 1, \pm 1)$ to the origin and the radius of the spheres, or $\sqrt{3} + 1$ . To verify this is the longest length, we can see from the triangle inequality that the length from the origin to any other point on the spheres is strictly smaller. Thus, the answer is $\boxed{\mathrm{D}}$ .

Solution 2

Simplify the problem - in $2$ dimensions, place $4$ circles in each quadrant in the coordinate plane. To find the circle enclosing these $4$ circles, notice that if you connect the $4$ centers as a square, the diameter of the large circle will be the diagonal of the square plus two small radii.

Similarly, in $3$ dimensions, consider the cube created by connecting the $8$ centers of the spheres. The diameter of the large spheres will just be the space diagonal of this cube, plus the two radiuses on either side. The side length of the cube is $2$ , so the space diagonal will be $2\sqrt{3}$ . Then the diameter of the large sphere is $2\sqrt{3} + 2$ and the radius is $\sqrt{3} + 1$ or $\boxed{\mathrm{D}}$ .

~Silverdragon

Video Solution by OmegaLearn

https://youtu.be/TT5UHS0H2Eo?t=1459

~ pi_is_3.14

@@ Line 4: / Line 4: @@
 <math>
-\mathrm (A)\ \sqrt{2}  \qquad
+\mathrm {(A)}\ \sqrt{2}  \qquad
-\mathrm (B)\ \sqrt{3}  \qquad
+\mathrm {(B)}\ \sqrt{3}  \qquad
-\mathrm (C)\ 1+\sqrt{2}\qquad
+\mathrm {(C)}\ 1+\sqrt{2}\qquad
-\mathrm (D)\ 1+\sqrt{3}\qquad
+\mathrm {(D)}\ 1+\sqrt{3}\qquad
-\mathrm (E)\ 3
+\mathrm {(E)}\ 3
 </math>
 == Solution ==
-The eight spheres are formed by shifting spheres of radius <math>2</math> and center <math>(0, 0, 0)</math> <math>\pm 1</math> in the <math>x, y, z</math> directions. Hence, the centers of the spheres are <math>(\pm 1, \pm 1, \pm 1)</math>. For a sphere to contain all eight spheres, its radius must be greater than or equal to the longest distance from the origin to one of these spheres. This length is the intersection of one of the spheres and a line that goes through the origin and that sphere's center. This length is the sum of the distance from <math>(\pm 1, \pm 1, \pm 1)</math> to the origin and the radius of the sphere, or <math>\sqrt{3} + 1</math>. To verify this is the longest length from the origin to a point on any of the spheres, we can see from the triangle inequality that the length from the origin to any other point on the sphere is strictly smaller.
+The eight spheres are formed by shifting spheres of radius <math>1</math> and center <math>(0, 0, 0)</math> <math>\pm 1</math> in the <math>x, y, z</math> directions. Hence, the centers of the spheres are <math>(\pm 1, \pm 1, \pm 1)</math>. For a sphere centered at the origin to contain all eight spheres, its radius must be greater than or equal to the longest distance from the origin to one of these spheres. This length is the sum of the distance from <math>(\pm 1, \pm 1, \pm 1)</math> to the origin and the radius of the spheres, or <math>\sqrt{3} + 1</math>. To verify this is the longest length, we can see from the triangle inequality that the length from the origin to any other point on the spheres is strictly smaller. Thus, the answer is <math>\boxed{\mathrm{D}}</math>.
+== Solution 2 ==
+Simplify the problem - in <math>2</math> dimensions, place <math>4</math> circles in each quadrant in the coordinate plane. To find the circle enclosing these <math>4</math> circles, notice that if you connect the <math>4</math> centers as a square, the diameter of the large circle will be the diagonal of the square plus two small radii.
+Similarly, in <math>3</math> dimensions, consider the cube created by connecting the <math>8</math> centers of the spheres. The diameter of the large spheres will just be the space diagonal of this cube, plus the two radiuses on either side. The side length of the cube is <math>2</math>, so the space diagonal will be <math>2\sqrt{3}</math>. Then the diameter of the large sphere is <math>2\sqrt{3} + 2</math> and the radius is <math>\sqrt{3} + 1</math> or <math>\boxed{\mathrm{D}}</math>.
+~Silverdragon
+== Video Solution by OmegaLearn ==
+https://youtu.be/TT5UHS0H2Eo?t=1459
+~ pi_is_3.14
 == See also ==
-* [[2005 AMC 12B Problems]]
+{{AMC12 box|year=2005|ab=B|num-b=15|num-a=17}}
+[[Category:Introductory Geometry Problems]]
+[[Category:3D Geometry Problems]]
+{{MAA Notice}}

2005 AMC 12B (Problems • Answer Key • Resources)
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