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2011 AMC 12A Problems/Problem 15

Revision as of 22:14, 5 January 2024 by Jaspersun (talk | contribs) (Added another solution.)
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Problem

The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?

$\textbf{(A)}\ 3\sqrt{2} \qquad \textbf{(B)}\ \frac{13}{3} \qquad \textbf{(C)}\ 4\sqrt{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{13}{2}$

Solution 1

Let $ABCDE$ be the pyramid with $ABCD$ as the square base. Let $O$ and $M$ be the center of square $ABCD$ and the midpoint of side $AB$ respectively. Lastly, let the hemisphere be tangent to the triangular face $ABE$ at $P$.

Notice that $\triangle EOM$ has a right angle at $O$. Since the hemisphere is tangent to the triangular face $ABE$ at $P$, $\angle EPO$ is also $90^{\circ}$. Hence $\triangle EOM$ is similar to $\triangle EPO$.

$\frac{OM}{2} = \frac{6}{EP}$

$OM = \frac{6}{EP} \times 2$

$OM = \frac{6}{\sqrt{6^2 - 2^2}} \times 2 = \frac{3\sqrt{2}}{2}$

The length of the square base is thus $2 \times \frac{3\sqrt{2}}{2} = 3\sqrt{2} \rightarrow \boxed{\textbf{A}}$

Solution 2 (Answer Choices)

Consider a cross section of the pyramid such that it is a hemisphere inscribed inside of a triangle. Let $A$ be the vertex furthest away from the hemisphere. Let $P$ be a point where the hemisphere is tangent. Let $O$ be the centre of the hemisphere. Triangle $AOP$ is a right triangle, since $P$ is ninety degrees. Apply the pythagorean formula to find side $AP$ as $4\sqrt{2}.$ Since we know the answer must have $\sqrt{2}$ somewhere, we can remove choices B, D and E from selection. Choice C is simply the length of $AP$, which cannot be the desired length. Thus, the answer must be $\boxed{\textbf{A}}$.

~ jaspersun

Video Solution

https://www.youtube.com/watch?v=u23iWcqbJlE ~Shreyas S

See also

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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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