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Difference between revisions of "2021 AMC 12B Problems/Problem 14"
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==Solution 1==
 
==Solution 1==
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This question is just about pythagorean theorem
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This question is just about Pythagorean theorem
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<cmath>a^2+(a+2)^2-b^2 = (a+4)^2</cmath>
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<cmath>\begin{align*}
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<cmath>2a^2+4a+4-b^2 = a^2+8a+16</cmath>
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a^2+(a+2)^2-b^2 &= (a+4)^2 \\
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<cmath>a^2-4a+4-b^2 = 16</cmath>
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2a^2+4a+4-b^2 &= a^2+8a+16 \\
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<cmath>(a-2+b)(a-2-b) = 16</cmath>
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a^2-4a+4-b^2 &= 16 \\
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<cmath>a=3, b=7</cmath>
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(a-2+b)(a-2-b) &= 16,
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With these calculation, we find out answer to be <math>\boxed{\textbf{(A) }24\sqrt5}</math> ~Lopkiloinm
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\end{align*}</cmath>
 +
from which <math>(a,b)=(3,7).</math> With these calculation, we find out answer to be <math>\boxed{\textbf{(A) }24\sqrt5}</math>
 +
 
 +
~Lopkiloinm
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==Solution 2==
 
==Solution 2==
 
Let <math>\overline{AD}</math> be <math>b</math>, <math>\overline{CD}</math> be <math>a</math>, <math>\overline{MD}</math> be <math>x</math>,  
 
Let <math>\overline{AD}</math> be <math>b</math>, <math>\overline{CD}</math> be <math>a</math>, <math>\overline{MD}</math> be <math>x</math>,  

Revision as of 07:24, 12 June 2021

Problem

Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$

$\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}$

Solution 1

This question is just about Pythagorean theorem \begin{align*} a^2+(a+2)^2-b^2 &= (a+4)^2 \\ 2a^2+4a+4-b^2 &= a^2+8a+16 \\ a^2-4a+4-b^2 &= 16 \\ (a-2+b)(a-2-b) &= 16, \end{align*} from which $(a,b)=(3,7).$ With these calculation, we find out answer to be $\boxed{\textbf{(A) }24\sqrt5}$

~Lopkiloinm

Solution 2

Let $\overline{AD}$ be $b$, $\overline{CD}$ be $a$, $\overline{MD}$ be $x$, $\overline{MC}$, $\overline{MA}$, $\overline{MB}$ be $t$, $t-2$, $t+2$ respectively.

We have three equations: \[a^2 + x^2 = t^2\] \[a^2 + b^2 + x^2 = t^2 + 4t + 4\] \[b^2 + x^2 = t^2 - 4t + 4\]

Subbing in the first and third equation into the second equation, we get: \[t^2 - 8t - x^2 = 0\] \[(t-4)^2 - x^2 = 16\] \[(t-4-x)(t-4+x) = 16\] Therefore, \[t = 9\], \[x = 3\] Solving for other values, we get $b = 2\sqrt{10}$, $a = 6\sqrt{2}$. The volume is then \[\frac{1}{3} abx = \boxed{\textbf{(A)}24\sqrt{5}}\] ~jamess2022(burntTacos)

Video Solution by Hawk Math

https://www.youtube.com/watch?v=p4iCAZRUESs

Video Solution by OmegaLearn (Pythagorean Theorem and Volume of Pyramid)

https://youtu.be/4_Oqp_ECLRw

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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