Difference between revisions of "2021 AMC 12B Problems/Problem 14"
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We have three equations: | We have three equations: | ||
− | <cmath>a^2 + x^2 = t^2 | + | <cmath>\begin{align*} |
− | + | a^2 + x^2 &= t^2, \ | |
− | <cmath> | + | a^2 + b^2 + x^2 &= t^2 + 4t + 4, \ |
+ | b^2 + x^2 &= t^2 - 4t + 4. | ||
+ | \end{align*}</cmath> | ||
+ | Subbing in the first and third equation into the second equation, we get: | ||
+ | <cmath>\begin{align} | ||
+ | t^2 - 8t - x^2 &= 0, \ | ||
+ | (t-4)^2 - x^2 &= 16, \ | ||
+ | (t-4-x)(t-4+x) &= 16. | ||
+ | \end{align*}</cmath> | ||
+ | Therefore, we have <math>t = 9</math> and <math>x = 3</math>. | ||
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− | |||
− | |||
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Solving for other values, we get <math>b = 2\sqrt{10}</math>, <math>a = 6\sqrt{2}</math>. | Solving for other values, we get <math>b = 2\sqrt{10}</math>, <math>a = 6\sqrt{2}</math>. | ||
− | The volume is then <cmath>\frac{1}{3} abx = \boxed{\textbf{(A)}24\sqrt{5}}</cmath> ~jamess2022(burntTacos) | + | The volume is then <cmath>\frac{1}{3} abx = \boxed{\textbf{(A)}24\sqrt{5}}</cmath> |
+ | |||
+ | ~jamess2022(burntTacos) | ||
==Video Solution by Hawk Math== | ==Video Solution by Hawk Math== |
Revision as of 06:31, 12 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
This question is just about Pythagorean theorem from which 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:
\begin{align} t^2 - 8t - x^2 &= 0, \\ (t-4)^2 - x^2 &= 16, \\ (t-4-x)(t-4+x) &= 16. \end{align*} (Error compiling LaTeX. Unknown error_msg)
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.