Difference between revisions of "1989 AHSME Problems/Problem 26"
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<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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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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<math>\frac{\frac{x^3}{6}}{x^3} \rightarrow \framebox[1.1\width]{(C) \frac{1}{6} }</math> | <math>\frac{\frac{x^3}{6}}{x^3} \rightarrow \framebox[1.1\width]{(C) \frac{1}{6} }</math> | ||
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Revision as of 20:27, 7 June 2014
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
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…
$\frac{\frac{x^3}{6}}{x^3} \rightarrow \framebox[1.1\width]{(C) \frac{1}{6} }$ (Error compiling LaTeX. Unknown error_msg)
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