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Difference between revisions of "2021 AMC 12B Problems/Problem 14"

(→‎Solution 2: Simplified the sentences using equal signs.)
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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

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

Let $MD=3$ and $MA=7.$ 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 $AD=b$, $CD=a$, $MD=x$, $MC=t$. It follows that $MA=t-2$ and $MB=t+2$.

We have three equations: \begin{align*} a^2 + x^2 &= t^2, \\ a^2 + b^2 + x^2 &= t^2 + 4t + 4, \\ b^2 + x^2 &= t^2 - 4t + 4. \end{align*} 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*} Therefore, we have $t = 9$ and $x = 3$.

Solving for other values, we get $b = 2\sqrt{10}$, $a = 6\sqrt{2}$. The volume is then \[\frac{1}{3} abx = \boxed{\textbf{(A) }24\sqrt5}.\]

~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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