2011 AMC 12B Problems/Problem 18

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?

$\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}$

Solution

We can use the Pythagorean Theorem to split one of the triangular faces into two 30-60-90 triangles with side lengths $\frac{1}{2}, 1$ and $\frac{\sqrt{3}}{2}$.

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 $\frac{\sqrt{3}}{2}$.

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 $\frac{1}{2}, \frac{\sqrt2}{2}$ and $\frac{\sqrt{3}}{2}$.

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 $x$, the small pyramid has height $\frac{x\sqrt{2}}{2}$ (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.

$x +\frac{x\sqrt{2}}{2} = \frac{\sqrt2}{2}$.

$x\left(1+\frac{\sqrt{2}}{2} \right) =\frac{\sqrt{2}}{2}$

$x\left(2+\sqrt{2}\right) = \sqrt{2}$

$x = \frac{\sqrt{2}}{2+\sqrt{2}}  \cdot \frac{2-\sqrt{2}}{2-\sqrt{2}} = \frac{2\sqrt{2}-2}{4-2} = \sqrt{2}-1 =$side length of cube.

$\left(\sqrt{2}-1\right)^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(-1) + 3(\sqrt{2})(-1)^2 + (-1)^3 = 2\sqrt{2} - 6 +3\sqrt{2} - 1 =\textbf{(A)} 5\sqrt{2} - 7$

(Faster Method)

We can arrive to the conclusion that the side length of the cube is $\sqrt{2}-1.$ Because this expression has a radical and an integer term we can expect a radical and an integer term in the answer choice. $\textbf{(A)} 5\sqrt{2} - 7$ is the only such choice.

See Also

2011 AMC 12B (ProblemsAnswer KeyResources)
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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