Difference between revisions of "2021 AMC 12B Problems/Problem 14"
MRENTHUSIASM (talk | contribs) (→Solution 2: Simplified the sentences using equal signs.) |
MRENTHUSIASM (talk | contribs) m (→Solution 2) |
||
Line 27: | Line 27: | ||
Subbing in the first and third equation into the second equation, we get: | Subbing in the first and third equation into the second equation, we get: | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | t^2 - 8t - x^2 &= 0 | + | t^2 - 8t - x^2 &= 0 \ |
− | (t-4)^2 - x^2 &= 16 | + | (t-4)^2 - x^2 &= 16 \ |
(t-4-x)(t-4+x) &= 16. | (t-4-x)(t-4+x) &= 16. | ||
\end{align*}</cmath> | \end{align*}</cmath> |
Revision as of 09:39, 20 June 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
Let and This question is just about Pythagorean theorem from which With these calculation, we find out answer to be .
~Lopkiloinm
Solution 2
Let , , , . It follows that and .
We have three equations: Subbing in the first and third equation into the second equation, we get: Therefore, we have and .
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.