Difference between revisions of "2023 AMC 10A Problems/Problem 18"
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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> | <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. | ||
| + | ~aiden22gao | ||
Revision as of 17:54, 9 November 2023
A rhombic dodecahedron is a solid with 
congruent rhombus faces. At every vertex, 
or 
edges meet, depending on the vertex. How many vertices have exactly edges meet?
someone else do latex pls
Solution 1
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. ~aiden22gao
