Difference between revisions of "2014 AMC 12A Problems/Problem 17"
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Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively. | Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively. |
Latest revision as of 09:08, 7 July 2020
Problem
A rectangular box contains a sphere of radius and eight smaller spheres of radius . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is ?
Solution 1
Let be the point in the same plane as the centers of the top spheres equidistant from said centers. Let be the analogous point for the bottom spheres, and let be the midpoint of and the center of the large sphere. Let and be the points at which line intersects the top of the box and the bottom, respectively.
Let be the center of any of the top spheres (you choose!). We have , and , so . Similarly, . and are clearly equal to the radius of the small spheres, . Thus the total height is , or .
(Solution by AwesomeToad)
Solution 2
Let be the center of the large sphere and be the center of any small sphere. Let be a vertex of the rectangular prism closest to point . Let be the point on the edge of the prism such that and are perpendicular. Let points and point lie on and respectively such that and are perpendicular at .
is the radii of the spheres, so . is the shortest length between the center of a small sphere and the edge of the prism, so . Similarly, . Since is a rectangle, . Since , . Then, . is the length from to the top of the prism or . Thus, . The prism is symmetrical, so
(Solution by BJHHar)
Solution 3
take a cross section and see that h is made up of two radii of the circle plus some radical expression. the only choice satisfying this condition is (a)
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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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