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Difference between revisions of "2007 AMC 12A Problems/Problem 20"

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== Problem ==
 
== Problem ==
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Corners are sliced off a [[unit cube]] so that the six [[face]]s each become regular [[octagon]]s. What is the total [[volume]] of the removed [[tetrahedra]]?
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Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
 
 
 
<math>\mathrm{(A)}\ \frac{5\sqrt{2}-7}{3}\qquad \mathrm{(B)}\ \frac{10-7\sqrt{2}}{3}\qquad \mathrm{(C)}\ \frac{3-2\sqrt{2}}{3}\qquad \mathrm{(D)}\ \frac{8\sqrt{2}-11}{3}\qquad \mathrm{(E)}\ \frac{6-4\sqrt{2}}{3}</math>
 
<math>\mathrm{(A)}\ \frac{5\sqrt{2}-7}{3}\qquad \mathrm{(B)}\ \frac{10-7\sqrt{2}}{3}\qquad \mathrm{(C)}\ \frac{3-2\sqrt{2}}{3}\qquad \mathrm{(D)}\ \frac{8\sqrt{2}-11}{3}\qquad \mathrm{(E)}\ \frac{6-4\sqrt{2}}{3}</math>
  
 
== Solution ==
 
== Solution ==
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{{image}}
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[[Image:2007 AMC 12A Problem 20.png|150px]]
  
 
Since the sides of a [[regular polygon]] are equal in length, we can call each side <math>x</math>. Examine one edge of the unit cube: each contains two slanted diagonal edges of an octagon and one straight edge. The diagonal edges form <math>45-45-90 \triangle</math> [[right triangle]]s, making the distance on the edge of the cube <math>\frac{x}{\sqrt{2}}</math>. Thus, <math>2 \cdot \frac{x}{\sqrt{2}} + x = 1</math>, and <math>x = \frac{1}{\sqrt{2} + 1} \cdot \left(\frac{\sqrt{2} - 1}{\sqrt{2} - 1}\right) = \sqrt{2} - 1</math>.  
 
Since the sides of a [[regular polygon]] are equal in length, we can call each side <math>x</math>. Examine one edge of the unit cube: each contains two slanted diagonal edges of an octagon and one straight edge. The diagonal edges form <math>45-45-90 \triangle</math> [[right triangle]]s, making the distance on the edge of the cube <math>\frac{x}{\sqrt{2}}</math>. Thus, <math>2 \cdot \frac{x}{\sqrt{2}} + x = 1</math>, and <math>x = \frac{1}{\sqrt{2} + 1} \cdot \left(\frac{\sqrt{2} - 1}{\sqrt{2} - 1}\right) = \sqrt{2} - 1</math>.  
  
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Each of the cut off corners is a [[pyramid]], whose volume can be calculated by <math>V = \frac 13 Bh</math>. Use the base as one of the three [[congruency (geometry)|congruent]] [[isosceles triangle]]s, with the height being one of the edges of the pyramid that sits on the edges of the cube. The height is <math>\frac x{\sqrt{2}} = 1 - \frac{1}{\sqrt{2}}</math>. The base is a <math>45-45-90 \triangle</math> with leg of length <math>1 - \frac{1}{\sqrt{2}}</math>, making its area <math>\frac 12\left(1 - \frac 1{\sqrt{2}}\right)^2 = \frac{3 - 2\sqrt{2}}4</math>. Plugging this in, we get that the area of one of the tetrahedra is <math>\frac 13 \left(1 - \frac{1}{\sqrt{2}}\right)\left(\frac{3 - 2\sqrt{2}}4\right) = \frac{10 - 7\sqrt{2}}{24}</math>. Since there are 8 removed corners, we get an answer of <math>\frac{10 - 7\sqrt{2}}{3} \Longrightarrow \mathrm{B}</math>
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Each of the cut off corners is a [[pyramid]], whose volume can be calculated by <math>V = \frac 13 Bh</math>. Use the base as one of the three [[congruency (geometry)|congruent]] [[isosceles triangle]]s, with the height being one of the edges of the pyramid that sits on the edges of the cube. The height is <math>\frac x{\sqrt{2}} = 1 - \frac{1}{\sqrt{2}}</math>. The base is a <math>45-45-90 \triangle</math> with leg of length <math>1 - \frac{1}{\sqrt{2}}</math>, making its area <math>\frac 12\left(1 - \frac 1{\sqrt{2}}\right)^2 = \frac{3 - 2\sqrt{2}}4</math>. Plugging this in, we get that the volume of one of the tetrahedra is <math>\frac 13 \left(1 - \frac{1}{\sqrt{2}}\right)\left(\frac{3 - 2\sqrt{2}}4\right) = \frac{10 - 7\sqrt{2}}{24}</math>. Since there are 8 removed corners, we get an answer of <math>\frac{10 - 7\sqrt{2}}{3} \Longrightarrow \boxed{\mathrm{B}}</math>
  
 
== See also ==
 
== See also ==
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[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 +
[[Category:3D Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 09:10, 16 September 2022

Problem

Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?

$\mathrm{(A)}\ \frac{5\sqrt{2}-7}{3}\qquad \mathrm{(B)}\ \frac{10-7\sqrt{2}}{3}\qquad \mathrm{(C)}\ \frac{3-2\sqrt{2}}{3}\qquad \mathrm{(D)}\ \frac{8\sqrt{2}-11}{3}\qquad \mathrm{(E)}\ \frac{6-4\sqrt{2}}{3}$

Solution

2007 AMC 12A Problem 20.png

Since the sides of a regular polygon are equal in length, we can call each side $x$. Examine one edge of the unit cube: each contains two slanted diagonal edges of an octagon and one straight edge. The diagonal edges form $45-45-90 \triangle$ right triangles, making the distance on the edge of the cube $\frac{x}{\sqrt{2}}$. Thus, $2 \cdot \frac{x}{\sqrt{2}} + x = 1$, and $x = \frac{1}{\sqrt{2} + 1} \cdot \left(\frac{\sqrt{2} - 1}{\sqrt{2} - 1}\right) = \sqrt{2} - 1$.

Each of the cut off corners is a pyramid, whose volume can be calculated by $V = \frac 13 Bh$. Use the base as one of the three congruent isosceles triangles, with the height being one of the edges of the pyramid that sits on the edges of the cube. The height is $\frac x{\sqrt{2}} = 1 - \frac{1}{\sqrt{2}}$. The base is a $45-45-90 \triangle$ with leg of length $1 - \frac{1}{\sqrt{2}}$, making its area $\frac 12\left(1 - \frac 1{\sqrt{2}}\right)^2 = \frac{3 - 2\sqrt{2}}4$. Plugging this in, we get that the volume of one of the tetrahedra is $\frac 13 \left(1 - \frac{1}{\sqrt{2}}\right)\left(\frac{3 - 2\sqrt{2}}4\right) = \frac{10 - 7\sqrt{2}}{24}$. Since there are 8 removed corners, we get an answer of $\frac{10 - 7\sqrt{2}}{3} \Longrightarrow \boxed{\mathrm{B}}$

See also

2007 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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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