Difference between revisions of "2021 AMC 12B Problems/Problem 14"
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==Problem== | ==Problem== | ||
− | Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math> | + | Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MABCD?</math> |
<math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math> | <math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math> |
Revision as of 19:35, 12 February 2021
Contents
[hide]Problem
Let be a rectangle and let be a segment perpendicular to the plane of . Suppose that has integer length, and the lengths of and are consecutive odd positive integers (in this order). What is the volume of pyramid
Solution 1
This question is just about pythagorean theorem With these calculation, we find out answer to be ~Lopkiloinm
Solution 2
Let be , be , be , , , be , , respectively.
We have three equations:
Subbing in the first and third equation into the second equation, we get: Therefore, , Solving for other values, we get , . The volume is then ~jamess2022(burntTacos)
Video Solution by Hawk Math
https://www.youtube.com/watch?v=p4iCAZRUESs
Video Solution by OmegaLearn (Pythagorean Theorem and Volume of Pyramid)
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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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