Difference between revisions of "1989 AHSME Problems/Problem 26"
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| + | == Problem == | ||
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A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is | A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is | ||
<math> \mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(E) \frac{1}{4} } </math> | <math> \mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(E) \frac{1}{4} } </math> | ||
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| + | == Solution == | ||
Call the length of a side of the cube x. Thus, the volume of the cube is <math>x^3</math>. We can then find that a side of this regular octahedron is the square root of <math>(\frac{x}{2})^2</math>+<math>(\frac{x}{2})^2</math> which is equivalent to <math>\frac{x\sqrt{2}}{2}</math>. Using our general formula for the volume of a regular octahedron of side length a, which is <math>\frac{a^3\sqrt2}{3}</math>, we get that the volume of this octahedron is... | Call the length of a side of the cube x. Thus, the volume of the cube is <math>x^3</math>. We can then find that a side of this regular octahedron is the square root of <math>(\frac{x}{2})^2</math>+<math>(\frac{x}{2})^2</math> which is equivalent to <math>\frac{x\sqrt{2}}{2}</math>. Using our general formula for the volume of a regular octahedron of side length a, which is <math>\frac{a^3\sqrt2}{3}</math>, we get that the volume of this octahedron is... | ||
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Comparing the ratio of the volume of the octahedron to the cube is… | Comparing the ratio of the volume of the octahedron to the cube is… | ||
| − | <math>\frac{\frac{x^3}{6}}{x^3} \rightarrow | + | <math>\frac{\frac{x^3}{6}}{x^3} \rightarrow \frac{1}{6} </math> or <math>\fbox{C}</math> |
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| + | == See also == | ||
| + | {{AHSME box|year=1989|num-b=25|num-a=27}} | ||
| + | |||
| + | [[Category: Intermediate Geometry Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 08:08, 22 October 2014
Problem
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
Solution
Call the length of a side of the cube x. Thus, the volume of the cube is 
. We can then find that a side of this regular octahedron is the square root of 
+ which is equivalent to 
. Using our general formula for the volume of a regular octahedron of side length a, which is 
, we get that the volume of this octahedron is...
Comparing the ratio of the volume of the octahedron to the cube is…
or 
See also
| 1989 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 25 |
Followed by Problem 27 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
