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Difference between revisions of "2023 AMC 10A Problems/Problem 18"

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A rhombic dodecahedron is a solid with <math>12</math> congruent rhombus faces. At every vertex, <math>3</math> or <math>4</math> edges meet, depending on the vertex. How many vertices have exactly <math>3</math> edges meet?
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<math>\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9</math>
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someone else do latex pls
 
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Solution 1
 
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Note Euler's formula where V+F-E=2. There are 12 faces and the number of edges is 24 because there are 12 faces each with four edges and each edge is shared by two faces. Now we know that there are 14 vertices on the figure. Let A be the number of vertices with degree 3 and B be the number of vertices with degree 4. A+B=14 is our first equation. Now note that the sum of the degrees of all the points is twice the number of edges. Now we know 3A+4B=48. Solving this system of equations gives B=6 and A = 8 so the answer is D.
 
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~aiden22gao
 

Latest revision as of 08:38, 14 October 2024

Explore Google, the most popular and widely used search engine globally, known for delivering fast, relevant, and reliable search results. In addition to search capabilities, Google offers a wide array of services and products, including Gmail for email, Google Maps for navigation, Google Drive for cloud storage, YouTube for videos, Google Photos for image management, and Google Docs for collaborative document editing. With a commitment to innovation, Google continuously improves its technologies to enhance your online experience across all devices, providing seamless integration with Android, Chrome, and more.

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Gmail: alaccary446@gmail.com

Hashtag: #Google trends #google news #google map #google drive

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