Difference between revisions of "2011 AMC 12B Problems/Problem 18"
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==Solution== | ==Solution== | ||
+ | ===Solution 1=== | ||
We can use the Pythagorean Theorem to split one of the triangular faces into two 30-60-90 triangles with side lengths <math>\frac{1}{2}, 1</math> and <math>\frac{\sqrt{3}}{2}</math>. | We can use the Pythagorean Theorem to split one of the triangular faces into two 30-60-90 triangles with side lengths <math>\frac{1}{2}, 1</math> and <math>\frac{\sqrt{3}}{2}</math>. | ||
− | Next, take a cross-section of the pyramid, forming a triangle with the top of the | + | Next, take a cross-section of the pyramid, forming a triangle with the top of the pyramid and the midpoints of two opposite sides of the square base. |
This triangle is isosceles with a base of 1 and two sides of length <math>\frac{\sqrt{3}}{2}</math>. | This triangle is isosceles with a base of 1 and two sides of length <math>\frac{\sqrt{3}}{2}</math>. | ||
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The height of this triangle will equal the height of the pyramid. To find this height, split the triangle into two right triangles, with sides <math>\frac{1}{2}, \frac{\sqrt2}{2}</math> and <math>\frac{\sqrt{3}}{2}</math>. | The height of this triangle will equal the height of the pyramid. To find this height, split the triangle into two right triangles, with sides <math>\frac{1}{2}, \frac{\sqrt2}{2}</math> and <math>\frac{\sqrt{3}}{2}</math>. | ||
− | The cube, touching all four triangular faces, will form a similar pyramid which sits on top of the cube. If the cube has side length <math>x</math>, the pyramid has | + | The cube, touching all four triangular faces, will form a similar pyramid which sits on top of the cube. If the cube has side length <math>x</math>, the small pyramid has height <math>\frac{x\sqrt{2}}{2}</math> (because the pyramids are similar). |
Thus, the height of the cube plus the height of the smaller pyramid equals the height of the larger pyramid. | Thus, the height of the cube plus the height of the smaller pyramid equals the height of the larger pyramid. | ||
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<math>\left(\sqrt{2}-1\right)^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(-1) + 3(\sqrt{2})(-1)^2 + (-1)^3 = 2\sqrt{2} - 6 +3\sqrt{2} - 1 =\textbf{(A)} 5\sqrt{2} - 7</math> | <math>\left(\sqrt{2}-1\right)^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(-1) + 3(\sqrt{2})(-1)^2 + (-1)^3 = 2\sqrt{2} - 6 +3\sqrt{2} - 1 =\textbf{(A)} 5\sqrt{2} - 7</math> | ||
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+ | ===Solution 2=== | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2011|num-b=17|num-a=19|ab=B}} | {{AMC12 box|year=2011|num-b=17|num-a=19|ab=B}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 01:07, 6 January 2018
Problem
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Solution
Solution 1
We can use the Pythagorean Theorem to split one of the triangular faces into two 30-60-90 triangles with side lengths and .
Next, take a cross-section of the pyramid, forming a triangle with the top of the pyramid and the midpoints of two opposite sides of the square base.
This triangle is isosceles with a base of 1 and two sides of length .
The height of this triangle will equal the height of the pyramid. To find this height, split the triangle into two right triangles, with sides and .
The cube, touching all four triangular faces, will form a similar pyramid which sits on top of the cube. If the cube has side length , the small pyramid has height (because the pyramids are similar).
Thus, the height of the cube plus the height of the smaller pyramid equals the height of the larger pyramid.
.
side length of cube.
Solution 2
See Also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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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